Polymath Wiki - User contributions [en]

Logo

2011-02-18T01:56:38Z

Thomas: added 3 more links

The Polymath wiki needs a logo.

Add links below to candidate images - it should be in a standard image file type, and available under a suitable copyright license.

== PNG format, Public domain ==

Three variations of a same idea: [http://thomas1111.files.wordpress.com/2011/02/logo1-ts.png quite formal], [http://thomas1111.files.wordpress.com/2011/02/logo2-ts.png over two lines],
[http://thomas1111.files.wordpress.com/2011/02/logo3-ts.png collaborative spirit].

Adding some more in response to comments: [http://thomas1111.files.wordpress.com/2011/02/logo14-ts.png map-like less pixelized], [http://thomas1111.files.wordpress.com/2011/02/logo16-ts1.png puzzle]

== JPEG format, Public domain ==

One with Da Vinci drawing in the background: [http://thomas1111.files.wordpress.com/2011/02/logo7-ts.jpg Vitruve man],

Logo

2011-02-17T19:08:21Z

Thomas: added section PNG/public domain, with three ideas.

"Low Dimensions" grant acknowledgments

2010-04-26T20:05:19Z

Thomas:

Participants should be arranged in alphabetical order of surname.

== Participants and contact information ==

(Note: this list is incomplete and unofficial. Inclusion or omission from this list should not be construed as any formal declaration of level of contribution to this project.)

* Kristal Cantwell
* Kareem Carr, NYU [http://twofoldgaze.wordpress.com/]
* Jason Dyer
* Kevin O'Bryant, CUNY (Staten Island and the Graduate Center), [http://www.math.csi.cuny.edu/obryant]
* Klas Markström, Umeå universitet, Sweden. [http://abel.math.umu.se/~klasm/]
* Michael Peake
* Terence Tao, UCLA, [http://www.math.ucla.edu/~tao]

== Grant information ==

* Kevin O'Bryant is supported by a grant from The City University of New York PSC-CUNY Research Award Program.
* Terence Tao is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.

== Other acknowledgments ==

Miscellaneous contributors to the project include KS Chua, Sune Kristian Jakobsen, Tyler Neylon (bounds and related quantities), Thomas Sauvaget.

Thanks to Michael Nielsen for hosting the polymath wiki for this project.

This project was a spinoff from the larger "Polymath1" project, initiated by Timothy Gowers.

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T05:03:36Z

Thomas: /* Analysis */

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884 this comment].

We notice that looking only at the first 246 elements of the sequence it already has reached C=3, so this shows (a) that the sequence is not the continuation of one of the longuest C=2 multiplicative sequences, (b) that it is possible to go on multiplicatively within the C=3 constraint for quite a long time indeed.

=The sequence=

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+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
---+--+-++--+--+-++-++--+++++-+----++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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+-++-++--+-++--+--+-++--+--+-++-++--+-++-+---+-++-++--+--+-+
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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++

=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.

So the backtracking has not retained all the initial sign flips. There are no further flips coming from primes equal to 2 mod 3.

Overall we thus have 31 sign flips, to be compared with the 1569 primes between 1 and 13186, that's a mere 1.97% of primes changed.

==Data==

prime, sign after flip

709 -

1201 -

1999 -

3457 -

3541 -

3613 -

4003 -

4219 -

5659 -

5953 -

6007 -

6163 -

6907 -

7297 -

7927 -

8803 -

9547 -

9619 -

9931 -

10039 -

10369 -

10993 -

11317 -

11443 -

11503 -

11887 -

12043 -

12049 -

12301 -

12763 -

12967 -

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T05:03:19Z

Thomas: /* Analysis */

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884 this comment].

We notice that looking only at the first 246 elements of the sequence it already has reached C=3, so this shows (a) that the sequence is not the continuation of one of the longuest C=2 multiplicative sequences, (b) that it is possible to go on multiplicatively within the C=3 constraint for quite a long time indeed.

=The sequence=

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+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
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+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
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+-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
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+-++--+--+-++--+--+-+++++--+-++-----+-++-+---+-+++++--+-++--
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+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
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+-++--+--+-++-++--+-++--+--+-++-++---+++-++--+-++--+--+-++--
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+-++--+-++--+--+-++-++--+-++-++--+-++--+----++--+--++++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++-+---+-++-++--+-++--
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+-++--+--++++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++++
+--+-+--++--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+--+-++--
+-++--+-+--+++-+--+-++-++--+-++--+--+-++-+---+-+--+++-+-++--
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+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
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---+--+-++--+--+-++-++--+++++-+----++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
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+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++

=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.

So the backtracking has not retained all the initial sign flips. There are no further flips coming from primes equal to 2 mod 3.

Overall we thus have 31 signs flips, to be compared with the 1569 primes between 1 and 13186, that's a mere 1.97% of primes changed.

==Data==

prime, sign after flip

709 -

1201 -

1999 -

3457 -

3541 -

3613 -

4003 -

4219 -

5659 -

5953 -

6007 -

6163 -

6907 -

7297 -

7927 -

8803 -

9547 -

9619 -

9931 -

10039 -

10369 -

10993 -

11317 -

11443 -

11503 -

11887 -

12043 -

12049 -

12301 -

12763 -

12967 -

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T04:51:54Z

Thomas:

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884 this comment].

We notice that looking only at the first 246 elements of the sequence it already has reached C=3, so this shows (a) that the sequence is not the continuation of one of the longuest C=2 multiplicative sequences, (b) that it is possible to go on multiplicatively within the C=3 constraint for quite a long time indeed.

=The sequence=

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
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+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
--++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.

So the backtracking has not retained all the initial sign flips. There are no further flips coming from primes equal to 2 mod 3.

==Data==

prime, sign after flip

709 -

1201 -

1999 -

3457 -

3541 -

3613 -

4003 -

4219 -

5659 -

5953 -

6007 -

6163 -

6907 -

7297 -

7927 -

8803 -

9547 -

9619 -

9931 -

10039 -

10369 -

10993 -

11317 -

11443 -

11503 -

11887 -

12043 -

12049 -

12301 -

12763 -

12967 -

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T04:51:18Z

Thomas: CORRECTED ERRORS!

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884 this comment].

We notice that looking only at the first 246 elements of the sequence it already has reached C=3, so this shows (a) that the sequence is not the continuation of one of the longuest C=2 multiplicative sequences, (b) that it is possible to go on multiplicatively within the C=3 constraint for quite a long time indeed.

=The sequence=

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+-++--+--+-++--+-++--+--+-++--+--++++-++--+-++--+--+-++-++--
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+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++----++-++-++--
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+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
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+-++--+--+-++-++--+-++--+--+-++-++---+++-++--+-++--+--+-++--
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+-++--+-++--+--+-++-++--+-++-++--+-++--+----++--+--++++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++-+---+-++-++--+-++--
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+-++--+--++++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++++
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+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+--+-++--
+-++--+-+--+++-+--+-++-++--+-++--+--+-++-+---+-+--+++-+-++--
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+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
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---+--+-++--+--+-++-++--+++++-+----++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
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+-++-++--+-++--+--+-++--+--+-++-++--+-++-+---+-++-++--+--+-+
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+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++

=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.

So the backtracking has not retained all the initial sign flips. There are no further flips coming from primes equal to 2 mod 3.

==Data==

prime, sign after flip

709 -
1201 -
1999 -
3457 -
3541 -
3613 -
4003 -
4219 -
5659 -
5953 -
6007 -
6163 -
6907 -
7297 -
7927 -
8803 -
9547 -
9619 -
9931 -
10039 -
10369 -
10993 -
11317 -
11443 -
11503 -
11887 -
12043 -
12049 -
12301 -
12763 -
12967 -

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T04:39:44Z

Thomas: more comments added

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884 this comment].

We notice that looking only at the first 246 elements of the sequence it already has reached C=3, so this shows (a) that the sequence is not the continuation of one of the longuest C=2 multiplicative sequences, (b) that it is possible to go on multiplicatively within the C=3 constraint for quite a long time indeed.

=The sequence=

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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
--++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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+-++--+-++--+--++++--+----++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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+-+---+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
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+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
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+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
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+-++--+-++-++--+-+---+--+-++-+++---++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
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+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
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+-+++---++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
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+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
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+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
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+-++--+--+-++--+--+-+++++--+-++-----+-++-+---+-+++++--+-++--
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+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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+-++--+--+-++--+-++--+--+-++--+--++++-++--+-++--+--+-++-++--
+-++-++--+-++--+----++-++--+-++-++--+-++--+--+-++--+--+-++-+
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+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
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+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++----++-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
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+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--++-
+-++-++--+--+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+-++-++--+--+--+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
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+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
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+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
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+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
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+-++--+--+-++-++--+-++--+--+-++-++---+++-++--+-++--+--+-++--
+----+++++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+----++--+--++++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++-+---+-++-++--+-++--
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+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
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+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--++++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++++
+--+-+--++--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+--+-++--
+-++--+-+--+++-+--+-++-++--+-++--+--+-++-+---+-+--+++-+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
---+--+-++--+--+-++-++--+++++-+----++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+-++-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++--+--+-++-++--+-++-+---+-++-++--+--+-+
+--+-++--+--+-++--+--+-++-++--+-++--+--++++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
+--+-+--++--+-++--+---+++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++

=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.
So the backtracking has not retained all the initial sign flips. There are many further flips, coming from primes equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) and have in fact been sent to +1.

Here are the primes where a sign flip from <math>\mu_3</math> occurs, with the new sign. There are 793 sign flips altogether, to be compared with the number of primes between 1 and 13186, namely 1569, so 50.54% of primes have been flipped.

==Data==

prime, sign after flip

2 +

5 +

11 +

17 +

23 +

29 +

41 +

47 +

53 +

59 +

71 +

83 +

89 +

101 +

107 +

113 +

131 +

137 +

149 +

167 +

173 +

179 +

191 +

197 +

227 +

233 +

239 +

251 +

257 +

263 +

269 +

281 +

293 +

311 +

317 +

347 +

353 +

359 +

383 +

389 +

401 +

419 +

431 +

443 +

449 +

461 +

467 +

479 +

491 +

503 +

509 +

521 +

557 +

563 +

569 +

587 +

593 +

599 +

617 +

641 +

647 +

653 +

659 +

677 +

683 +

701 +

709 -

719 +

743 +

761 +

773 +

797 +

809 +

821 +

827 +

839 +

857 +

863 +

887 +

911 +

929 +

941 +

947 +

953 +

971 +

977 +

983 +

1013 +

1019 +

1031 +

1049 +

1061 +

1091 +

1097 +

1103 +

1109 +

1151 +

1163 +

1181 +

1187 +

1193 +

1201 -

1217 +

1223 +

1229 +

1259 +

1277 +

1283 +

1289 +

1301 +

1307 +

1319 +

1361 +

1367 +

1373 +

1409 +

1427 +

1433 +

1439 +

1451 +

1481 +

1487 +

1493 +

1499 +

1511 +

1523 +

1553 +

1559 +

1571 +

1583 +

1601 +

1607 +

1613 +

1619 +

1637 +

1667 +

1709 +

1721 +

1733 +

1787 +

1811 +

1823 +

1847 +

1871 +

1877 +

1889 +

1901 +

1907 +

1913 +

1931 +

1949 +

1973 +

1979 +

1997 +

1999 -

2003 +

2027 +

2039 +

2063 +

2069 +

2081 +

2087 +

2099 +

2111 +

2129 +

2141 +

2153 +

2207 +

2213 +

2237 +

2243 +

2267 +

2273 +

2297 +

2309 +

2333 +

2339 +

2351 +

2357 +

2381 +

2393 +

2399 +

2411 +

2417 +

2423 +

2441 +

2447 +

2459 +

2477 +

2531 +

2543 +

2549 +

2579 +

2591 +

2609 +

2621 +

2633 +

2657 +

2663 +

2687 +

2693 +

2699 +

2711 +

2729 +

2741 +

2753 +

2777 +

2789 +

2801 +

2819 +

2843 +

2861 +

2879 +

2897 +

2903 +

2909 +

2927 +

2939 +

2957 +

2963 +

2969 +

2999 +

3011 +

3023 +

3041 +

3083 +

3089 +

3119 +

3137 +

3167 +

3191 +

3203 +

3209 +

3221 +

3251 +

3257 +

3299 +

3329 +

3347 +

3359 +

3371 +

3389 +

3407 +

3413 +

3449 +

3457 -

3461 +

3467 +

3491 +

3527 +

3533 +

3539 +

3541 -

3557 +

3581 +

3593 +

3613 -

3617 +

3623 +

3659 +

3671 +

3677 +

3701 +

3719 +

3761 +

3767 +

3779 +

3797 +

3803 +

3821 +

3833 +

3851 +

3863 +

3881 +

3911 +

3923 +

3929 +

3947 +

3989 +

4001 +

4003 -

4007 +

4013 +

4019 +

4049 +

4073 +

4079 +

4091 +

4127 +

4133 +

4139 +

4157 +

4211 +

4217 +

4219 -

4229 +

4241 +

4253 +

4271 +

4283 +

4289 +

4337 +

4349 +

4373 +

4391 +

4409 +

4421 +

4451 +

4457 +

4463 +

4481 +

4493 +

4517 +

4523 +

4547 +

4583 +

4637 +

4643 +

4649 +

4673 +

4679 +

4691 +

4703 +

4721 +

4733 +

4751 +

4787 +

4793 +

4817 +

4871 +

4877 +

4889 +

4919 +

4931 +

4937 +

4943 +

4967 +

4973 +

5003 +

5009 +

5021 +

5039 +

5081 +

5087 +

5099 +

5147 +

5153 +

5171 +

5189 +

5231 +

5237 +

5261 +

5273 +

5279 +

5297 +

5303 +

5309 +

5333 +

5351 +

5381 +

5387 +

5393 +

5399 +

5417 +

5441 +

5471 +

5477 +

5483 +

5501 +

5507 +

5519 +

5531 +

5573 +

5591 +

5639 +

5651 +

5657 +

5659 -

5669 +

5693 +

5711 +

5717 +

5741 +

5783 +

5801 +

5807 +

5813 +

5843 +

5849 +

5867 +

5879 +

5897 +

5903 +

5927 +

5939 +

5953 -

5981 +

5987 +

6007 -

6011 +

6029 +

6047 +

6053 +

6089 +

6101 +

6113 +

6131 +

6143 +

6163 -

6173 +

6197 +

6203 +

6221 +

6257 +

6263 +

6269 +

6287 +

6299 +

6311 +

6317 +

6323 +

6329 +

6353 +

6359 +

6389 +

6449 +

6473 +

6521 +

6551 +

6563 +

6569 +

6581 +

6599 +

6653 +

6659 +

6689 +

6701 +

6719 +

6737 +

6761 +

6779 +

6791 +

6803 +

6827 +

6833 +

6857 +

6863 +

6869 +

6899 +

6907 -

6911 +

6917 +

6947 +

6959 +

6971 +

6977 +

6983 +

7001 +

7013 +

7019 +

7043 +

7079 +

7103 +

7109 +

7121 +

7127 +

7151 +

7187 +

7193 +

7229 +

7247 +

7253 +

7283 +

7297 -

7307 +

7331 +

7349 +

7433 +

7451 +

7457 +

7481 +

7487 +

7499 +

7517 +

7523 +

7529 +

7541 +

7547 +

7559 +

7577 +

7583 +

7589 +

7607 +

7643 +

7649 +

7673 +

7691 +

7703 +

7727 +

7757 +

7793 +

7817 +

7823 +

7829 +

7841 +

7853 +

7877 +

7883 +

7901 +

7907 +

7919 +

7927 -

7937 +

7949 +

8009 +

8039 +

8069 +

8081 +

8087 +

8093 +

8111 +

8117 +

8123 +

8147 +

8171 +

8219 +

8231 +

8237 +

8243 +

8273 +

8297 +

8363 +

8369 +

8387 +

8423 +

8429 +

8447 +

8501 +

8537 +

8543 +

8573 +

8597 +

8609 +

8627 +

8663 +

8669 +

8681 +

8693 +

8699 +

8741 +

8747 +

8803 -

8807 +

8819 +

8831 +

8837 +

8849 +

8861 +

8867 +

8933 +

8951 +

8963 +

8969 +

8999 +

9011 +

9029 +

9041 +

9059 +

9137 +

9161 +

9173 +

9203 +

9209 +

9221 +

9239 +

9257 +

9281 +

9293 +

9311 +

9323 +

9341 +

9371 +

9377 +

9413 +

9419 +

9431 +

9437 +

9461 +

9467 +

9473 +

9479 +

9491 +

9497 +

9521 +

9533 +

9539 +

9547 -

9551 +

9619 -

9623 +

9629 +

9677 +

9719 +

9743 +

9767 +

9791 +

9803 +

9833 +

9839 +

9851 +

9857 +

9887 +

9923 +

9929 +

9931 -

9941 +

10007 +

10037 +

10039 -

10061 +

10067 +

10079 +

10091 +

10133 +

10139 +

10151 +

10163 +

10169 +

10181 +

10193 +

10211 +

10223 +

10247 +

10253 +

10259 +

10271 +

10289 +

10301 +

10313 +

10331 +

10337 +

10343 +

10369 -

10391 +

10427 +

10433 +

10457 +

10463 +

10487 +

10499 +

10529 +

10589 +

10601 +

10607 +

10613 +

10631 +

10667 +

10691 +

10709 +

10733 +

10739 +

10781 +

10799 +

10847 +

10853 +

10859 +

10883 +

10889 +

10937 +

10949 +

10973 +

10979 +

10993 -

11003 +

11027 +

11057 +

11069 +

11087 +

11093 +

11117 +

11159 +

11171 +

11213 +

11243 +

11261 +

11273 +

11279 +

11317 -

11321 +

11351 +

11369 +

11393 +

11399 +

11411 +

11423 +

11443 -

11447 +

11471 +

11483 +

11489 +

11503 -

11519 +

11549 +

11579 +

11597 +

11621 +

11633 +

11657 +

11681 +

11699 +

11777 +

11783 +

11789 +

11801 +

11807 +

11813 +

11867 +

11887 -

11897 +

11903 +

11909 +

11927 +

11933 +

11939 +

11969 +

11981 +

11987 +

12011 +

12041 +

12043 -

12049 -

12071 +

12101 +

12107 +

12113 +

12119 +

12143 +

12149 +

12161 +

12197 +

12203 +

12227 +

12239 +

12251 +

12263 +

12269 +

12281 +

12301 -

12323 +

12347 +

12377 +

12401 +

12413 +

12437 +

12473 +

12479 +

12491 +

12497 +

12503 +

12527 +

12539 +

12569 +

12611 +

12641 +

12647 +

12653 +

12659 +

12671 +

12689 +

12713 +

12743 +

12763 -

12791 +

12809 +

12893 +

12899 +

12911 +

12917 +

12923 +

12941 +

12953 +

12959 +

12967 -

13001 +

13007 +

13037 +

13043 +

13049 +

13103 +

13109 +

13121 +

13127 +

13163 +

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T03:59:07Z

Thomas:

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884 this comment].

=The sequence=

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+--++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-+++-+--+-++--+--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
--++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--++++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--++++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-+--++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-+---+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+-++-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
++++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+--+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++---+-+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--++-+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++-----+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-+---+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++--+--+-++-++--++++--+--+-++-++--
--+++++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-+
+-++-++--+----++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
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+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--++-+-+--++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++-++--+-+--++--+-++--+--+-++--+--+-++-++--+--+--++-
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--++++--+----++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--++-
+-+---+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+----++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++--+--+--+--+-+
+-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+-++-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-+---+--+-++-+++---++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++--++-+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++----++--+--+-++-++--+-++-++--+-++--+--+-++--+-++-+
+-+++---++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++-++--+-+--+++-+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-+++-+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++-++----++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-+++-+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+----++--++-+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++-+---+-++-++--+-++--+--+-++--+--+-+
++++--+-+---+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+--+--++-+-++-++--+-+++-+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-+--++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+++
+--+--+-++-++--+-++--+--+-++-++----++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
+--+-+--++--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++----++-++--++++--+--+-++--+--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-+---+-++--+--+-++--+--+-++-++--++++--+--+-++--+--+-++-+
+--+-++--+--+-++-++--+-+--++--+-++--+--+-++-++-++--++++----+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--++++-++--
+-++--+--+-++--+--+-+++++--+-++-----+-++-+---+-+++++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-+---+-++--++-+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
+--+-+---+--+-++-++--+-++-++--+-++--+--+-++--++-+-++-++----+
+--+--++++-++--+-+--++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+++--++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++--++-+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--+--+-++-++--+-+++-+----++-++--+-++-++--+-++--+--+-+
+-++--+--+-++--+-++--+--+-++--+--++++-++--+-++--+--+-++-++--
+-++-++--+-++--+----++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-++---+++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++-++-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++----++-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--++-
+-++-++--+--+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+-++-++--+--+--+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++--+--+-++--+--+-++-++--+-++-++-++-++--+--+-++--+--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++----++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++---+++-++--+-++--+--+-++--
+----+++++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+----++--+--++++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++-+---+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-+++++----++-++--+-++--+--+-++-++--+--+-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--++++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++++
+--+-+--++--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+--+-++--
+-++--+-+--+++-+--+-++-++--+-++--+--+-++-+---+-+--+++-+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
---+--+-++--+--+-++-++--+++++-+----++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+-++-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++--+--+-++-++--+-++-+---+-++-++--+--+-+
+--+-++--+--+-++--+--+-++-++--+-++--+--++++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
+--+-+--++--+-++--+---+++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++

=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.
So the backtracking has not retained all the initial sign flips. There are many further flips, coming from primes equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) and have in fact been sent to +1.

Here are the primes where a sign flip from <math>\mu_3</math> occurs, with the new sign. There are 793 sign flips altogether, to be compared with the number of primes between 1 and 13186, namely 1569, so 50.54% of primes have been flipped.

==Data==

prime, sign after flip

2 +

5 +

11 +

17 +

23 +

29 +

41 +

47 +

53 +

59 +

71 +

83 +

89 +

101 +

107 +

113 +

131 +

137 +

149 +

167 +

173 +

179 +

191 +

197 +

227 +

233 +

239 +

251 +

257 +

263 +

269 +

281 +

293 +

311 +

317 +

347 +

353 +

359 +

383 +

389 +

401 +

419 +

431 +

443 +

449 +

461 +

467 +

479 +

491 +

503 +

509 +

521 +

557 +

563 +

569 +

587 +

593 +

599 +

617 +

641 +

647 +

653 +

659 +

677 +

683 +

701 +

709 -

719 +

743 +

761 +

773 +

797 +

809 +

821 +

827 +

839 +

857 +

863 +

887 +

911 +

929 +

941 +

947 +

953 +

971 +

977 +

983 +

1013 +

1019 +

1031 +

1049 +

1061 +

1091 +

1097 +

1103 +

1109 +

1151 +

1163 +

1181 +

1187 +

1193 +

1201 -

1217 +

1223 +

1229 +

1259 +

1277 +

1283 +

1289 +

1301 +

1307 +

1319 +

1361 +

1367 +

1373 +

1409 +

1427 +

1433 +

1439 +

1451 +

1481 +

1487 +

1493 +

1499 +

1511 +

1523 +

1553 +

1559 +

1571 +

1583 +

1601 +

1607 +

1613 +

1619 +

1637 +

1667 +

1709 +

1721 +

1733 +

1787 +

1811 +

1823 +

1847 +

1871 +

1877 +

1889 +

1901 +

1907 +

1913 +

1931 +

1949 +

1973 +

1979 +

1997 +

1999 -

2003 +

2027 +

2039 +

2063 +

2069 +

2081 +

2087 +

2099 +

2111 +

2129 +

2141 +

2153 +

2207 +

2213 +

2237 +

2243 +

2267 +

2273 +

2297 +

2309 +

2333 +

2339 +

2351 +

2357 +

2381 +

2393 +

2399 +

2411 +

2417 +

2423 +

2441 +

2447 +

2459 +

2477 +

2531 +

2543 +

2549 +

2579 +

2591 +

2609 +

2621 +

2633 +

2657 +

2663 +

2687 +

2693 +

2699 +

2711 +

2729 +

2741 +

2753 +

2777 +

2789 +

2801 +

2819 +

2843 +

2861 +

2879 +

2897 +

2903 +

2909 +

2927 +

2939 +

2957 +

2963 +

2969 +

2999 +

3011 +

3023 +

3041 +

3083 +

3089 +

3119 +

3137 +

3167 +

3191 +

3203 +

3209 +

3221 +

3251 +

3257 +

3299 +

3329 +

3347 +

3359 +

3371 +

3389 +

3407 +

3413 +

3449 +

3457 -

3461 +

3467 +

3491 +

3527 +

3533 +

3539 +

3541 -

3557 +

3581 +

3593 +

3613 -

3617 +

3623 +

3659 +

3671 +

3677 +

3701 +

3719 +

3761 +

3767 +

3779 +

3797 +

3803 +

3821 +

3833 +

3851 +

3863 +

3881 +

3911 +

3923 +

3929 +

3947 +

3989 +

4001 +

4003 -

4007 +

4013 +

4019 +

4049 +

4073 +

4079 +

4091 +

4127 +

4133 +

4139 +

4157 +

4211 +

4217 +

4219 -

4229 +

4241 +

4253 +

4271 +

4283 +

4289 +

4337 +

4349 +

4373 +

4391 +

4409 +

4421 +

4451 +

4457 +

4463 +

4481 +

4493 +

4517 +

4523 +

4547 +

4583 +

4637 +

4643 +

4649 +

4673 +

4679 +

4691 +

4703 +

4721 +

4733 +

4751 +

4787 +

4793 +

4817 +

4871 +

4877 +

4889 +

4919 +

4931 +

4937 +

4943 +

4967 +

4973 +

5003 +

5009 +

5021 +

5039 +

5081 +

5087 +

5099 +

5147 +

5153 +

5171 +

5189 +

5231 +

5237 +

5261 +

5273 +

5279 +

5297 +

5303 +

5309 +

5333 +

5351 +

5381 +

5387 +

5393 +

5399 +

5417 +

5441 +

5471 +

5477 +

5483 +

5501 +

5507 +

5519 +

5531 +

5573 +

5591 +

5639 +

5651 +

5657 +

5659 -

5669 +

5693 +

5711 +

5717 +

5741 +

5783 +

5801 +

5807 +

5813 +

5843 +

5849 +

5867 +

5879 +

5897 +

5903 +

5927 +

5939 +

5953 -

5981 +

5987 +

6007 -

6011 +

6029 +

6047 +

6053 +

6089 +

6101 +

6113 +

6131 +

6143 +

6163 -

6173 +

6197 +

6203 +

6221 +

6257 +

6263 +

6269 +

6287 +

6299 +

6311 +

6317 +

6323 +

6329 +

6353 +

6359 +

6389 +

6449 +

6473 +

6521 +

6551 +

6563 +

6569 +

6581 +

6599 +

6653 +

6659 +

6689 +

6701 +

6719 +

6737 +

6761 +

6779 +

6791 +

6803 +

6827 +

6833 +

6857 +

6863 +

6869 +

6899 +

6907 -

6911 +

6917 +

6947 +

6959 +

6971 +

6977 +

6983 +

7001 +

7013 +

7019 +

7043 +

7079 +

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7187 +

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7229 +

7247 +

7253 +

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7433 +

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7487 +

7499 +

7517 +

7523 +

7529 +

7541 +

7547 +

7559 +

7577 +

7583 +

7589 +

7607 +

7643 +

7649 +

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7691 +

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7727 +

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7793 +

7817 +

7823 +

7829 +

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7877 +

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7901 +

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7919 +

7927 -

7937 +

7949 +

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8039 +

8069 +

8081 +

8087 +

8093 +

8111 +

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8219 +

8231 +

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8369 +

8387 +

8423 +

8429 +

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8501 +

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8543 +

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8627 +

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8741 +

8747 +

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8819 +

8831 +

8837 +

8849 +

8861 +

8867 +

8933 +

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9029 +

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9203 +

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9281 +

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9311 +

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9413 +

9419 +

9431 +

9437 +

9461 +

9467 +

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9491 +

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9521 +

9533 +

9539 +

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9619 -

9623 +

9629 +

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9719 +

9743 +

9767 +

9791 +

9803 +

9833 +

9839 +

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9923 +

9929 +

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9941 +

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10037 +

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10061 +

10067 +

10079 +

10091 +

10133 +

10139 +

10151 +

10163 +

10169 +

10181 +

10193 +

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10247 +

10253 +

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10301 +

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10427 +

10433 +

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10487 +

10499 +

10529 +

10589 +

10601 +

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10631 +

10667 +

10691 +

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10733 +

10739 +

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10847 +

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10889 +

10937 +

10949 +

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10979 +

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11003 +

11027 +

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11069 +

11087 +

11093 +

11117 +

11159 +

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11213 +

11243 +

11261 +

11273 +

11279 +

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11321 +

11351 +

11369 +

11393 +

11399 +

11411 +

11423 +

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11447 +

11471 +

11483 +

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11519 +

11549 +

11579 +

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11621 +

11633 +

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11681 +

11699 +

11777 +

11783 +

11789 +

11801 +

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11813 +

11867 +

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11897 +

11903 +

11909 +

11927 +

11933 +

11939 +

11969 +

11981 +

11987 +

12011 +

12041 +

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12071 +

12101 +

12107 +

12113 +

12119 +

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12149 +

12161 +

12197 +

12203 +

12227 +

12239 +

12251 +

12263 +

12269 +

12281 +

12301 -

12323 +

12347 +

12377 +

12401 +

12413 +

12437 +

12473 +

12479 +

12491 +

12497 +

12503 +

12527 +

12539 +

12569 +

12611 +

12641 +

12647 +

12653 +

12659 +

12671 +

12689 +

12713 +

12743 +

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12791 +

12809 +

12893 +

12899 +

12911 +

12917 +

12923 +

12941 +

12953 +

12959 +

12967 -

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13037 +

13043 +

13049 +

13103 +

13109 +

13121 +

13127 +

13163 +

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T03:49:02Z

Thomas:

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884 this comment].

=The sequence=

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+-++-++--+--+--+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++--+--+-++--+--+-++-++--+-++-++-++-++--+--+-++--+--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++----++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++---+++-++--+-++--+--+-++--
+----+++++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+----++--+--++++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++-+---+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-+++++----++-++--+-++--+--+-++-++--+--+-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--++++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++++
+--+-+--++--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+--+-++--
+-++--+-+--+++-+--+-++-++--+-++--+--+-++-+---+-+--+++-+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
---+--+-++--+--+-++-++--+++++-+----++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+-++-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++--+--+-++-++--+-++-+---+-++-++--+--+-+
+--+-++--+--+-++--+--+-++-++--+-++--+--++++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
+--+-+--++--+-++--+---+++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++

=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.
So the backtracking has not retained all the initial sign flips. There are many further flips, coming from primes equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) and have in fact been sent to +1.

Here are the primes where a sign flip from <math>\mu_3</math> occurs, with the new sign. There are 793 sign flips altogether, to be compared with the number of primes between 1 and 13186, namely 1569, so 50.54% of primes have been flipped.

==Data==

prime sign after flip
2 +

5 +

11 +

17 +

23 +

29 +

41 +

47 +

53 +

59 +

71 +

83 +

89 +

101 +

107 +

113 +

131 +

137 +

149 +

167 +

173 +

179 +

191 +

197 +

227 +

233 +

239 +

251 +

257 +

263 +

269 +

281 +

293 +

311 +

317 +

347 +

353 +

359 +

383 +

389 +

401 +

419 +

431 +

443 +

449 +

461 +

467 +

479 +

491 +

503 +

509 +

521 +

557 +

563 +

569 +

587 +

593 +

599 +

617 +

641 +

647 +

653 +

659 +

677 +

683 +

701 +

709 -

719 +

743 +

761 +

773 +

797 +

809 +

821 +

827 +

839 +

857 +

863 +

887 +

911 +

929 +

941 +

947 +

953 +

971 +

977 +

983 +

1013 +

1019 +

1031 +

1049 +

1061 +

1091 +

1097 +

1103 +

1109 +

1151 +

1163 +

1181 +

1187 +

1193 +

1201 -

1217 +

1223 +

1229 +

1259 +

1277 +

1283 +

1289 +

1301 +

1307 +

1319 +

1361 +

1367 +

1373 +

1409 +

1427 +

1433 +

1439 +

1451 +

1481 +

1487 +

1493 +

1499 +

1511 +

1523 +

1553 +

1559 +

1571 +

1583 +

1601 +

1607 +

1613 +

1619 +

1637 +

1667 +

1709 +

1721 +

1733 +

1787 +

1811 +

1823 +

1847 +

1871 +

1877 +

1889 +

1901 +

1907 +

1913 +

1931 +

1949 +

1973 +

1979 +

1997 +

1999 -

2003 +

2027 +

2039 +

2063 +

2069 +

2081 +

2087 +

2099 +

2111 +

2129 +

2141 +

2153 +

2207 +

2213 +

2237 +

2243 +

2267 +

2273 +

2297 +

2309 +

2333 +

2339 +

2351 +

2357 +

2381 +

2393 +

2399 +

2411 +

2417 +

2423 +

2441 +

2447 +

2459 +

2477 +

2531 +

2543 +

2549 +

2579 +

2591 +

2609 +

2621 +

2633 +

2657 +

2663 +

2687 +

2693 +

2699 +

2711 +

2729 +

2741 +

2753 +

2777 +

2789 +

2801 +

2819 +

2843 +

2861 +

2879 +

2897 +

2903 +

2909 +

2927 +

2939 +

2957 +

2963 +

2969 +

2999 +

3011 +

3023 +

3041 +

3083 +

3089 +

3119 +

3137 +

3167 +

3191 +

3203 +

3209 +

3221 +

3251 +

3257 +

3299 +

3329 +

3347 +

3359 +

3371 +

3389 +

3407 +

3413 +

3449 +

3457 -

3461 +

3467 +

3491 +

3527 +

3533 +

3539 +

3541 -

3557 +

3581 +

3593 +

3613 -

3617 +

3623 +

3659 +

3671 +

3677 +

3701 +

3719 +

3761 +

3767 +

3779 +

3797 +

3803 +

3821 +

3833 +

3851 +

3863 +

3881 +

3911 +

3923 +

3929 +

3947 +

3989 +

4001 +

4003 -

4007 +

4013 +

4019 +

4049 +

4073 +

4079 +

4091 +

4127 +

4133 +

4139 +

4157 +

4211 +

4217 +

4219 -

4229 +

4241 +

4253 +

4271 +

4283 +

4289 +

4337 +

4349 +

4373 +

4391 +

4409 +

4421 +

4451 +

4457 +

4463 +

4481 +

4493 +

4517 +

4523 +

4547 +

4583 +

4637 +

4643 +

4649 +

4673 +

4679 +

4691 +

4703 +

4721 +

4733 +

4751 +

4787 +

4793 +

4817 +

4871 +

4877 +

4889 +

4919 +

4931 +

4937 +

4943 +

4967 +

4973 +

5003 +

5009 +

5021 +

5039 +

5081 +

5087 +

5099 +

5147 +

5153 +

5171 +

5189 +

5231 +

5237 +

5261 +

5273 +

5279 +

5297 +

5303 +

5309 +

5333 +

5351 +

5381 +

5387 +

5393 +

5399 +

5417 +

5441 +

5471 +

5477 +

5483 +

5501 +

5507 +

5519 +

5531 +

5573 +

5591 +

5639 +

5651 +

5657 +

5659 -

5669 +

5693 +

5711 +

5717 +

5741 +

5783 +

5801 +

5807 +

5813 +

5843 +

5849 +

5867 +

5879 +

5897 +

5903 +

5927 +

5939 +

5953 -

5981 +

5987 +

6007 -

6011 +

6029 +

6047 +

6053 +

6089 +

6101 +

6113 +

6131 +

6143 +

6163 -

6173 +

6197 +

6203 +

6221 +

6257 +

6263 +

6269 +

6287 +

6299 +

6311 +

6317 +

6323 +

6329 +

6353 +

6359 +

6389 +

6449 +

6473 +

6521 +

6551 +

6563 +

6569 +

6581 +

6599 +

6653 +

6659 +

6689 +

6701 +

6719 +

6737 +

6761 +

6779 +

6791 +

6803 +

6827 +

6833 +

6857 +

6863 +

6869 +

6899 +

6907 -

6911 +

6917 +

6947 +

6959 +

6971 +

6977 +

6983 +

7001 +

7013 +

7019 +

7043 +

7079 +

7103 +

7109 +

7121 +

7127 +

7151 +

7187 +

7193 +

7229 +

7247 +

7253 +

7283 +

7297 -

7307 +

7331 +

7349 +

7433 +

7451 +

7457 +

7481 +

7487 +

7499 +

7517 +

7523 +

7529 +

7541 +

7547 +

7559 +

7577 +

7583 +

7589 +

7607 +

7643 +

7649 +

7673 +

7691 +

7703 +

7727 +

7757 +

7793 +

7817 +

7823 +

7829 +

7841 +

7853 +

7877 +

7883 +

7901 +

7907 +

7919 +

7927 -

7937 +

7949 +

8009 +

8039 +

8069 +

8081 +

8087 +

8093 +

8111 +

8117 +

8123 +

8147 +

8171 +

8219 +

8231 +

8237 +

8243 +

8273 +

8297 +

8363 +

8369 +

8387 +

8423 +

8429 +

8447 +

8501 +

8537 +

8543 +

8573 +

8597 +

8609 +

8627 +

8663 +

8669 +

8681 +

8693 +

8699 +

8741 +

8747 +

8803 -

8807 +

8819 +

8831 +

8837 +

8849 +

8861 +

8867 +

8933 +

8951 +

8963 +

8969 +

8999 +

9011 +

9029 +

9041 +

9059 +

9137 +

9161 +

9173 +

9203 +

9209 +

9221 +

9239 +

9257 +

9281 +

9293 +

9311 +

9323 +

9341 +

9371 +

9377 +

9413 +

9419 +

9431 +

9437 +

9461 +

9467 +

9473 +

9479 +

9491 +

9497 +

9521 +

9533 +

9539 +

9547 -

9551 +

9619 -

9623 +

9629 +

9677 +

9719 +

9743 +

9767 +

9791 +

9803 +

9833 +

9839 +

9851 +

9857 +

9887 +

9923 +

9929 +

9931 -

9941 +

10007 +

10037 +

10039 -

10061 +

10067 +

10079 +

10091 +

10133 +

10139 +

10151 +

10163 +

10169 +

10181 +

10193 +

10211 +

10223 +

10247 +

10253 +

10259 +

10271 +

10289 +

10301 +

10313 +

10331 +

10337 +

10343 +

10369 -

10391 +

10427 +

10433 +

10457 +

10463 +

10487 +

10499 +

10529 +

10589 +

10601 +

10607 +

10613 +

10631 +

10667 +

10691 +

10709 +

10733 +

10739 +

10781 +

10799 +

10847 +

10853 +

10859 +

10883 +

10889 +

10937 +

10949 +

10973 +

10979 +

10993 -

11003 +

11027 +

11057 +

11069 +

11087 +

11093 +

11117 +

11159 +

11171 +

11213 +

11243 +

11261 +

11273 +

11279 +

11317 -

11321 +

11351 +

11369 +

11393 +

11399 +

11411 +

11423 +

11443 -

11447 +

11471 +

11483 +

11489 +

11503 -

11519 +

11549 +

11579 +

11597 +

11621 +

11633 +

11657 +

11681 +

11699 +

11777 +

11783 +

11789 +

11801 +

11807 +

11813 +

11867 +

11887 -

11897 +

11903 +

11909 +

11927 +

11933 +

11939 +

11969 +

11981 +

11987 +

12011 +

12041 +

12043 -

12049 -

12071 +

12101 +

12107 +

12113 +

12119 +

12143 +

12149 +

12161 +

12197 +

12203 +

12227 +

12239 +

12251 +

12263 +

12269 +

12281 +

12301 -

12323 +

12347 +

12377 +

12401 +

12413 +

12437 +

12473 +

12479 +

12491 +

12497 +

12503 +

12527 +

12539 +

12569 +

12611 +

12641 +

12647 +

12653 +

12659 +

12671 +

12689 +

12713 +

12743 +

12763 -

12791 +

12809 +

12893 +

12899 +

12911 +

12917 +

12923 +

12941 +

12953 +

12959 +

12967 -

13001 +

13007 +

13037 +

13043 +

13049 +

13103 +

13109 +

13121 +

13127 +

13163 +

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T03:47:08Z

Thomas:

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884 this comment].

=The sequence=

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=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.
So the backtracking has not retained all the initial sign flips. There are many further flips, coming from primes equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) and have in fact been sent to +1.

Here are the primes where a sign flip from <math\mu_3</math> occurs, with the new sign. There are 793 sign flips altogether, to be compared with the number of primes between 1 and 13186, namely 1569, so 50.54% of primes have been flipped.

==Data==

prime sign after flip
2 +

5 +

11 +

17 +

23 +

29 +

41 +

47 +

53 +

59 +

71 +

83 +

89 +

101 +

107 +

113 +

131 +

137 +

149 +

167 +

173 +

179 +

191 +

197 +

227 +

233 +

239 +

251 +

257 +

263 +

269 +

281 +

293 +

311 +

317 +

347 +

353 +

359 +

383 +

389 +

401 +

419 +

431 +

443 +

449 +

461 +

467 +

479 +

491 +

503 +

509 +

521 +

557 +

563 +

569 +

587 +

593 +

599 +

617 +

641 +

647 +

653 +

659 +

677 +

683 +

701 +

709 -

719 +

743 +

761 +

773 +

797 +

809 +

821 +

827 +

839 +

857 +

863 +

887 +

911 +

929 +

941 +

947 +

953 +

971 +

977 +

983 +

1013 +

1019 +

1031 +

1049 +

1061 +

1091 +

1097 +

1103 +

1109 +

1151 +

1163 +

1181 +

1187 +

1193 +

1201 -

1217 +

1223 +

1229 +

1259 +

1277 +

1283 +

1289 +

1301 +

1307 +

1319 +

1361 +

1367 +

1373 +

1409 +

1427 +

1433 +

1439 +

1451 +

1481 +

1487 +

1493 +

1499 +

1511 +

1523 +

1553 +

1559 +

1571 +

1583 +

1601 +

1607 +

1613 +

1619 +

1637 +

1667 +

1709 +

1721 +

1733 +

1787 +

1811 +

1823 +

1847 +

1871 +

1877 +

1889 +

1901 +

1907 +

1913 +

1931 +

1949 +

1973 +

1979 +

1997 +

1999 -

2003 +

2027 +

2039 +

2063 +

2069 +

2081 +

2087 +

2099 +

2111 +

2129 +

2141 +

2153 +

2207 +

2213 +

2237 +

2243 +

2267 +

2273 +

2297 +

2309 +

2333 +

2339 +

2351 +

2357 +

2381 +

2393 +

2399 +

2411 +

2417 +

2423 +

2441 +

2447 +

2459 +

2477 +

2531 +

2543 +

2549 +

2579 +

2591 +

2609 +

2621 +

2633 +

2657 +

2663 +

2687 +

2693 +

2699 +

2711 +

2729 +

2741 +

2753 +

2777 +

2789 +

2801 +

2819 +

2843 +

2861 +

2879 +

2897 +

2903 +

2909 +

2927 +

2939 +

2957 +

2963 +

2969 +

2999 +

3011 +

3023 +

3041 +

3083 +

3089 +

3119 +

3137 +

3167 +

3191 +

3203 +

3209 +

3221 +

3251 +

3257 +

3299 +

3329 +

3347 +

3359 +

3371 +

3389 +

3407 +

3413 +

3449 +

3457 -

3461 +

3467 +

3491 +

3527 +

3533 +

3539 +

3541 -

3557 +

3581 +

3593 +

3613 -

3617 +

3623 +

3659 +

3671 +

3677 +

3701 +

3719 +

3761 +

3767 +

3779 +

3797 +

3803 +

3821 +

3833 +

3851 +

3863 +

3881 +

3911 +

3923 +

3929 +

3947 +

3989 +

4001 +

4003 -

4007 +

4013 +

4019 +

4049 +

4073 +

4079 +

4091 +

4127 +

4133 +

4139 +

4157 +

4211 +

4217 +

4219 -

4229 +

4241 +

4253 +

4271 +

4283 +

4289 +

4337 +

4349 +

4373 +

4391 +

4409 +

4421 +

4451 +

4457 +

4463 +

4481 +

4493 +

4517 +

4523 +

4547 +

4583 +

4637 +

4643 +

4649 +

4673 +

4679 +

4691 +

4703 +

4721 +

4733 +

4751 +

4787 +

4793 +

4817 +

4871 +

4877 +

4889 +

4919 +

4931 +

4937 +

4943 +

4967 +

4973 +

5003 +

5009 +

5021 +

5039 +

5081 +

5087 +

5099 +

5147 +

5153 +

5171 +

5189 +

5231 +

5237 +

5261 +

5273 +

5279 +

5297 +

5303 +

5309 +

5333 +

5351 +

5381 +

5387 +

5393 +

5399 +

5417 +

5441 +

5471 +

5477 +

5483 +

5501 +

5507 +

5519 +

5531 +

5573 +

5591 +

5639 +

5651 +

5657 +

5659 -

5669 +

5693 +

5711 +

5717 +

5741 +

5783 +

5801 +

5807 +

5813 +

5843 +

5849 +

5867 +

5879 +

5897 +

5903 +

5927 +

5939 +

5953 -

5981 +

5987 +

6007 -

6011 +

6029 +

6047 +

6053 +

6089 +

6101 +

6113 +

6131 +

6143 +

6163 -

6173 +

6197 +

6203 +

6221 +

6257 +

6263 +

6269 +

6287 +

6299 +

6311 +

6317 +

6323 +

6329 +

6353 +

6359 +

6389 +

6449 +

6473 +

6521 +

6551 +

6563 +

6569 +

6581 +

6599 +

6653 +

6659 +

6689 +

6701 +

6719 +

6737 +

6761 +

6779 +

6791 +

6803 +

6827 +

6833 +

6857 +

6863 +

6869 +

6899 +

6907 -

6911 +

6917 +

6947 +

6959 +

6971 +

6977 +

6983 +

7001 +

7013 +

7019 +

7043 +

7079 +

7103 +

7109 +

7121 +

7127 +

7151 +

7187 +

7193 +

7229 +

7247 +

7253 +

7283 +

7297 -

7307 +

7331 +

7349 +

7433 +

7451 +

7457 +

7481 +

7487 +

7499 +

7517 +

7523 +

7529 +

7541 +

7547 +

7559 +

7577 +

7583 +

7589 +

7607 +

7643 +

7649 +

7673 +

7691 +

7703 +

7727 +

7757 +

7793 +

7817 +

7823 +

7829 +

7841 +

7853 +

7877 +

7883 +

7901 +

7907 +

7919 +

7927 -

7937 +

7949 +

8009 +

8039 +

8069 +

8081 +

8087 +

8093 +

8111 +

8117 +

8123 +

8147 +

8171 +

8219 +

8231 +

8237 +

8243 +

8273 +

8297 +

8363 +

8369 +

8387 +

8423 +

8429 +

8447 +

8501 +

8537 +

8543 +

8573 +

8597 +

8609 +

8627 +

8663 +

8669 +

8681 +

8693 +

8699 +

8741 +

8747 +

8803 -

8807 +

8819 +

8831 +

8837 +

8849 +

8861 +

8867 +

8933 +

8951 +

8963 +

8969 +

8999 +

9011 +

9029 +

9041 +

9059 +

9137 +

9161 +

9173 +

9203 +

9209 +

9221 +

9239 +

9257 +

9281 +

9293 +

9311 +

9323 +

9341 +

9371 +

9377 +

9413 +

9419 +

9431 +

9437 +

9461 +

9467 +

9473 +

9479 +

9491 +

9497 +

9521 +

9533 +

9539 +

9547 -

9551 +

9619 -

9623 +

9629 +

9677 +

9719 +

9743 +

9767 +

9791 +

9803 +

9833 +

9839 +

9851 +

9857 +

9887 +

9923 +

9929 +

9931 -

9941 +

10007 +

10037 +

10039 -

10061 +

10067 +

10079 +

10091 +

10133 +

10139 +

10151 +

10163 +

10169 +

10181 +

10193 +

10211 +

10223 +

10247 +

10253 +

10259 +

10271 +

10289 +

10301 +

10313 +

10331 +

10337 +

10343 +

10369 -

10391 +

10427 +

10433 +

10457 +

10463 +

10487 +

10499 +

10529 +

10589 +

10601 +

10607 +

10613 +

10631 +

10667 +

10691 +

10709 +

10733 +

10739 +

10781 +

10799 +

10847 +

10853 +

10859 +

10883 +

10889 +

10937 +

10949 +

10973 +

10979 +

10993 -

11003 +

11027 +

11057 +

11069 +

11087 +

11093 +

11117 +

11159 +

11171 +

11213 +

11243 +

11261 +

11273 +

11279 +

11317 -

11321 +

11351 +

11369 +

11393 +

11399 +

11411 +

11423 +

11443 -

11447 +

11471 +

11483 +

11489 +

11503 -

11519 +

11549 +

11579 +

11597 +

11621 +

11633 +

11657 +

11681 +

11699 +

11777 +

11783 +

11789 +

11801 +

11807 +

11813 +

11867 +

11887 -

11897 +

11903 +

11909 +

11927 +

11933 +

11939 +

11969 +

11981 +

11987 +

12011 +

12041 +

12043 -

12049 -

12071 +

12101 +

12107 +

12113 +

12119 +

12143 +

12149 +

12161 +

12197 +

12203 +

12227 +

12239 +

12251 +

12263 +

12269 +

12281 +

12301 -

12323 +

12347 +

12377 +

12401 +

12413 +

12437 +

12473 +

12479 +

12491 +

12497 +

12503 +

12527 +

12539 +

12569 +

12611 +

12641 +

12647 +

12653 +

12659 +

12671 +

12689 +

12713 +

12743 +

12763 -

12791 +

12809 +

12893 +

12899 +

12911 +

12917 +

12923 +

12941 +

12953 +

12959 +

12967 -

13001 +

13007 +

13037 +

13043 +

13049 +

13103 +

13109 +

13121 +

13127 +

13163 +

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T03:46:26Z

Thomas:

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [[http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884] this comment].

=The sequence=

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+--++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-+++-+--+-++--+--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
--++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--++++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--++++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-+--++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-+---+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+-++-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
++++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
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=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

==Analysis==

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.
So the backtracking has not retained all the initial sign flips. There are many further flips, coming from primes equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) and have in fact been sent to +1.

Here are the primes where a sign flip from <math\mu_3</math> occurs, with the new sign. There are 793 sign flips altogether, to be compared with the number of primes between 1 and 13186, namely 1569, so 50.54% of primes have been flipped.

==Data==

prime sign after flip
2 +

5 +

11 +

17 +

23 +

29 +

41 +

47 +

53 +

59 +

71 +

83 +

89 +

101 +

107 +

113 +

131 +

137 +

149 +

167 +

173 +

179 +

191 +

197 +

227 +

233 +

239 +

251 +

257 +

263 +

269 +

281 +

293 +

311 +

317 +

347 +

353 +

359 +

383 +

389 +

401 +

419 +

431 +

443 +

449 +

461 +

467 +

479 +

491 +

503 +

509 +

521 +

557 +

563 +

569 +

587 +

593 +

599 +

617 +

641 +

647 +

653 +

659 +

677 +

683 +

701 +

709 -

719 +

743 +

761 +

773 +

797 +

809 +

821 +

827 +

839 +

857 +

863 +

887 +

911 +

929 +

941 +

947 +

953 +

971 +

977 +

983 +

1013 +

1019 +

1031 +

1049 +

1061 +

1091 +

1097 +

1103 +

1109 +

1151 +

1163 +

1181 +

1187 +

1193 +

1201 -

1217 +

1223 +

1229 +

1259 +

1277 +

1283 +

1289 +

1301 +

1307 +

1319 +

1361 +

1367 +

1373 +

1409 +

1427 +

1433 +

1439 +

1451 +

1481 +

1487 +

1493 +

1499 +

1511 +

1523 +

1553 +

1559 +

1571 +

1583 +

1601 +

1607 +

1613 +

1619 +

1637 +

1667 +

1709 +

1721 +

1733 +

1787 +

1811 +

1823 +

1847 +

1871 +

1877 +

1889 +

1901 +

1907 +

1913 +

1931 +

1949 +

1973 +

1979 +

1997 +

1999 -

2003 +

2027 +

2039 +

2063 +

2069 +

2081 +

2087 +

2099 +

2111 +

2129 +

2141 +

2153 +

2207 +

2213 +

2237 +

2243 +

2267 +

2273 +

2297 +

2309 +

2333 +

2339 +

2351 +

2357 +

2381 +

2393 +

2399 +

2411 +

2417 +

2423 +

2441 +

2447 +

2459 +

2477 +

2531 +

2543 +

2549 +

2579 +

2591 +

2609 +

2621 +

2633 +

2657 +

2663 +

2687 +

2693 +

2699 +

2711 +

2729 +

2741 +

2753 +

2777 +

2789 +

2801 +

2819 +

2843 +

2861 +

2879 +

2897 +

2903 +

2909 +

2927 +

2939 +

2957 +

2963 +

2969 +

2999 +

3011 +

3023 +

3041 +

3083 +

3089 +

3119 +

3137 +

3167 +

3191 +

3203 +

3209 +

3221 +

3251 +

3257 +

3299 +

3329 +

3347 +

3359 +

3371 +

3389 +

3407 +

3413 +

3449 +

3457 -

3461 +

3467 +

3491 +

3527 +

3533 +

3539 +

3541 -

3557 +

3581 +

3593 +

3613 -

3617 +

3623 +

3659 +

3671 +

3677 +

3701 +

3719 +

3761 +

3767 +

3779 +

3797 +

3803 +

3821 +

3833 +

3851 +

3863 +

3881 +

3911 +

3923 +

3929 +

3947 +

3989 +

4001 +

4003 -

4007 +

4013 +

4019 +

4049 +

4073 +

4079 +

4091 +

4127 +

4133 +

4139 +

4157 +

4211 +

4217 +

4219 -

4229 +

4241 +

4253 +

4271 +

4283 +

4289 +

4337 +

4349 +

4373 +

4391 +

4409 +

4421 +

4451 +

4457 +

4463 +

4481 +

4493 +

4517 +

4523 +

4547 +

4583 +

4637 +

4643 +

4649 +

4673 +

4679 +

4691 +

4703 +

4721 +

4733 +

4751 +

4787 +

4793 +

4817 +

4871 +

4877 +

4889 +

4919 +

4931 +

4937 +

4943 +

4967 +

4973 +

5003 +

5009 +

5021 +

5039 +

5081 +

5087 +

5099 +

5147 +

5153 +

5171 +

5189 +

5231 +

5237 +

5261 +

5273 +

5279 +

5297 +

5303 +

5309 +

5333 +

5351 +

5381 +

5387 +

5393 +

5399 +

5417 +

5441 +

5471 +

5477 +

5483 +

5501 +

5507 +

5519 +

5531 +

5573 +

5591 +

5639 +

5651 +

5657 +

5659 -

5669 +

5693 +

5711 +

5717 +

5741 +

5783 +

5801 +

5807 +

5813 +

5843 +

5849 +

5867 +

5879 +

5897 +

5903 +

5927 +

5939 +

5953 -

5981 +

5987 +

6007 -

6011 +

6029 +

6047 +

6053 +

6089 +

6101 +

6113 +

6131 +

6143 +

6163 -

6173 +

6197 +

6203 +

6221 +

6257 +

6263 +

6269 +

6287 +

6299 +

6311 +

6317 +

6323 +

6329 +

6353 +

6359 +

6389 +

6449 +

6473 +

6521 +

6551 +

6563 +

6569 +

6581 +

6599 +

6653 +

6659 +

6689 +

6701 +

6719 +

6737 +

6761 +

6779 +

6791 +

6803 +

6827 +

6833 +

6857 +

6863 +

6869 +

6899 +

6907 -

6911 +

6917 +

6947 +

6959 +

6971 +

6977 +

6983 +

7001 +

7013 +

7019 +

7043 +

7079 +

7103 +

7109 +

7121 +

7127 +

7151 +

7187 +

7193 +

7229 +

7247 +

7253 +

7283 +

7297 -

7307 +

7331 +

7349 +

7433 +

7451 +

7457 +

7481 +

7487 +

7499 +

7517 +

7523 +

7529 +

7541 +

7547 +

7559 +

7577 +

7583 +

7589 +

7607 +

7643 +

7649 +

7673 +

7691 +

7703 +

7727 +

7757 +

7793 +

7817 +

7823 +

7829 +

7841 +

7853 +

7877 +

7883 +

7901 +

7907 +

7919 +

7927 -

7937 +

7949 +

8009 +

8039 +

8069 +

8081 +

8087 +

8093 +

8111 +

8117 +

8123 +

8147 +

8171 +

8219 +

8231 +

8237 +

8243 +

8273 +

8297 +

8363 +

8369 +

8387 +

8423 +

8429 +

8447 +

8501 +

8537 +

8543 +

8573 +

8597 +

8609 +

8627 +

8663 +

8669 +

8681 +

8693 +

8699 +

8741 +

8747 +

8803 -

8807 +

8819 +

8831 +

8837 +

8849 +

8861 +

8867 +

8933 +

8951 +

8963 +

8969 +

8999 +

9011 +

9029 +

9041 +

9059 +

9137 +

9161 +

9173 +

9203 +

9209 +

9221 +

9239 +

9257 +

9281 +

9293 +

9311 +

9323 +

9341 +

9371 +

9377 +

9413 +

9419 +

9431 +

9437 +

9461 +

9467 +

9473 +

9479 +

9491 +

9497 +

9521 +

9533 +

9539 +

9547 -

9551 +

9619 -

9623 +

9629 +

9677 +

9719 +

9743 +

9767 +

9791 +

9803 +

9833 +

9839 +

9851 +

9857 +

9887 +

9923 +

9929 +

9931 -

9941 +

10007 +

10037 +

10039 -

10061 +

10067 +

10079 +

10091 +

10133 +

10139 +

10151 +

10163 +

10169 +

10181 +

10193 +

10211 +

10223 +

10247 +

10253 +

10259 +

10271 +

10289 +

10301 +

10313 +

10331 +

10337 +

10343 +

10369 -

10391 +

10427 +

10433 +

10457 +

10463 +

10487 +

10499 +

10529 +

10589 +

10601 +

10607 +

10613 +

10631 +

10667 +

10691 +

10709 +

10733 +

10739 +

10781 +

10799 +

10847 +

10853 +

10859 +

10883 +

10889 +

10937 +

10949 +

10973 +

10979 +

10993 -

11003 +

11027 +

11057 +

11069 +

11087 +

11093 +

11117 +

11159 +

11171 +

11213 +

11243 +

11261 +

11273 +

11279 +

11317 -

11321 +

11351 +

11369 +

11393 +

11399 +

11411 +

11423 +

11443 -

11447 +

11471 +

11483 +

11489 +

11503 -

11519 +

11549 +

11579 +

11597 +

11621 +

11633 +

11657 +

11681 +

11699 +

11777 +

11783 +

11789 +

11801 +

11807 +

11813 +

11867 +

11887 -

11897 +

11903 +

11909 +

11927 +

11933 +

11939 +

11969 +

11981 +

11987 +

12011 +

12041 +

12043 -

12049 -

12071 +

12101 +

12107 +

12113 +

12119 +

12143 +

12149 +

12161 +

12197 +

12203 +

12227 +

12239 +

12251 +

12263 +

12269 +

12281 +

12301 -

12323 +

12347 +

12377 +

12401 +

12413 +

12437 +

12473 +

12479 +

12491 +

12497 +

12503 +

12527 +

12539 +

12569 +

12611 +

12641 +

12647 +

12653 +

12659 +

12671 +

12689 +

12713 +

12743 +

12763 -

12791 +

12809 +

12893 +

12899 +

12911 +

12917 +

12923 +

12941 +

12953 +

12959 +

12967 -

13001 +

13007 +

13037 +

13043 +

13049 +

13103 +

13109 +

13121 +

13127 +

13163 +

Multiplicative sequences

2010-02-06T03:44:30Z

Thomas:

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T03:43:56Z

Thomas: more comments and structure

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [[http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884] this comment].

=The sequence=

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+--++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-+++-+--+-++--+--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
--++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--++++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--++++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-+--++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-+---+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+-++-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
++++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
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=Primes sent to -1=

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

=Sign flips=

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.
So the backtracking has not retained all the initial sign flips. There are many further flips, coming from primes equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) and have in fact been sent to +1.

Here are the primes where a sign flip from <math\mu_3</math> occurs, with the new sign. There are 793 sign flips altogether, to be compared with the number of primes between 1 and 13186, namely 1569, so 50.54% of primes have been flipped.

prime sign after flip
2 +

5 +

11 +

17 +

23 +

29 +

41 +

47 +

53 +

59 +

71 +

83 +

89 +

101 +

107 +

113 +

131 +

137 +

149 +

167 +

173 +

179 +

191 +

197 +

227 +

233 +

239 +

251 +

257 +

263 +

269 +

281 +

293 +

311 +

317 +

347 +

353 +

359 +

383 +

389 +

401 +

419 +

431 +

443 +

449 +

461 +

467 +

479 +

491 +

503 +

509 +

521 +

557 +

563 +

569 +

587 +

593 +

599 +

617 +

641 +

647 +

653 +

659 +

677 +

683 +

701 +

709 -

719 +

743 +

761 +

773 +

797 +

809 +

821 +

827 +

839 +

857 +

863 +

887 +

911 +

929 +

941 +

947 +

953 +

971 +

977 +

983 +

1013 +

1019 +

1031 +

1049 +

1061 +

1091 +

1097 +

1103 +

1109 +

1151 +

1163 +

1181 +

1187 +

1193 +

1201 -

1217 +

1223 +

1229 +

1259 +

1277 +

1283 +

1289 +

1301 +

1307 +

1319 +

1361 +

1367 +

1373 +

1409 +

1427 +

1433 +

1439 +

1451 +

1481 +

1487 +

1493 +

1499 +

1511 +

1523 +

1553 +

1559 +

1571 +

1583 +

1601 +

1607 +

1613 +

1619 +

1637 +

1667 +

1709 +

1721 +

1733 +

1787 +

1811 +

1823 +

1847 +

1871 +

1877 +

1889 +

1901 +

1907 +

1913 +

1931 +

1949 +

1973 +

1979 +

1997 +

1999 -

2003 +

2027 +

2039 +

2063 +

2069 +

2081 +

2087 +

2099 +

2111 +

2129 +

2141 +

2153 +

2207 +

2213 +

2237 +

2243 +

2267 +

2273 +

2297 +

2309 +

2333 +

2339 +

2351 +

2357 +

2381 +

2393 +

2399 +

2411 +

2417 +

2423 +

2441 +

2447 +

2459 +

2477 +

2531 +

2543 +

2549 +

2579 +

2591 +

2609 +

2621 +

2633 +

2657 +

2663 +

2687 +

2693 +

2699 +

2711 +

2729 +

2741 +

2753 +

2777 +

2789 +

2801 +

2819 +

2843 +

2861 +

2879 +

2897 +

2903 +

2909 +

2927 +

2939 +

2957 +

2963 +

2969 +

2999 +

3011 +

3023 +

3041 +

3083 +

3089 +

3119 +

3137 +

3167 +

3191 +

3203 +

3209 +

3221 +

3251 +

3257 +

3299 +

3329 +

3347 +

3359 +

3371 +

3389 +

3407 +

3413 +

3449 +

3457 -

3461 +

3467 +

3491 +

3527 +

3533 +

3539 +

3541 -

3557 +

3581 +

3593 +

3613 -

3617 +

3623 +

3659 +

3671 +

3677 +

3701 +

3719 +

3761 +

3767 +

3779 +

3797 +

3803 +

3821 +

3833 +

3851 +

3863 +

3881 +

3911 +

3923 +

3929 +

3947 +

3989 +

4001 +

4003 -

4007 +

4013 +

4019 +

4049 +

4073 +

4079 +

4091 +

4127 +

4133 +

4139 +

4157 +

4211 +

4217 +

4219 -

4229 +

4241 +

4253 +

4271 +

4283 +

4289 +

4337 +

4349 +

4373 +

4391 +

4409 +

4421 +

4451 +

4457 +

4463 +

4481 +

4493 +

4517 +

4523 +

4547 +

4583 +

4637 +

4643 +

4649 +

4673 +

4679 +

4691 +

4703 +

4721 +

4733 +

4751 +

4787 +

4793 +

4817 +

4871 +

4877 +

4889 +

4919 +

4931 +

4937 +

4943 +

4967 +

4973 +

5003 +

5009 +

5021 +

5039 +

5081 +

5087 +

5099 +

5147 +

5153 +

5171 +

5189 +

5231 +

5237 +

5261 +

5273 +

5279 +

5297 +

5303 +

5309 +

5333 +

5351 +

5381 +

5387 +

5393 +

5399 +

5417 +

5441 +

5471 +

5477 +

5483 +

5501 +

5507 +

5519 +

5531 +

5573 +

5591 +

5639 +

5651 +

5657 +

5659 -

5669 +

5693 +

5711 +

5717 +

5741 +

5783 +

5801 +

5807 +

5813 +

5843 +

5849 +

5867 +

5879 +

5897 +

5903 +

5927 +

5939 +

5953 -

5981 +

5987 +

6007 -

6011 +

6029 +

6047 +

6053 +

6089 +

6101 +

6113 +

6131 +

6143 +

6163 -

6173 +

6197 +

6203 +

6221 +

6257 +

6263 +

6269 +

6287 +

6299 +

6311 +

6317 +

6323 +

6329 +

6353 +

6359 +

6389 +

6449 +

6473 +

6521 +

6551 +

6563 +

6569 +

6581 +

6599 +

6653 +

6659 +

6689 +

6701 +

6719 +

6737 +

6761 +

6779 +

6791 +

6803 +

6827 +

6833 +

6857 +

6863 +

6869 +

6899 +

6907 -

6911 +

6917 +

6947 +

6959 +

6971 +

6977 +

6983 +

7001 +

7013 +

7019 +

7043 +

7079 +

7103 +

7109 +

7121 +

7127 +

7151 +

7187 +

7193 +

7229 +

7247 +

7253 +

7283 +

7297 -

7307 +

7331 +

7349 +

7433 +

7451 +

7457 +

7481 +

7487 +

7499 +

7517 +

7523 +

7529 +

7541 +

7547 +

7559 +

7577 +

7583 +

7589 +

7607 +

7643 +

7649 +

7673 +

7691 +

7703 +

7727 +

7757 +

7793 +

7817 +

7823 +

7829 +

7841 +

7853 +

7877 +

7883 +

7901 +

7907 +

7919 +

7927 -

7937 +

7949 +

8009 +

8039 +

8069 +

8081 +

8087 +

8093 +

8111 +

8117 +

8123 +

8147 +

8171 +

8219 +

8231 +

8237 +

8243 +

8273 +

8297 +

8363 +

8369 +

8387 +

8423 +

8429 +

8447 +

8501 +

8537 +

8543 +

8573 +

8597 +

8609 +

8627 +

8663 +

8669 +

8681 +

8693 +

8699 +

8741 +

8747 +

8803 -

8807 +

8819 +

8831 +

8837 +

8849 +

8861 +

8867 +

8933 +

8951 +

8963 +

8969 +

8999 +

9011 +

9029 +

9041 +

9059 +

9137 +

9161 +

9173 +

9203 +

9209 +

9221 +

9239 +

9257 +

9281 +

9293 +

9311 +

9323 +

9341 +

9371 +

9377 +

9413 +

9419 +

9431 +

9437 +

9461 +

9467 +

9473 +

9479 +

9491 +

9497 +

9521 +

9533 +

9539 +

9547 -

9551 +

9619 -

9623 +

9629 +

9677 +

9719 +

9743 +

9767 +

9791 +

9803 +

9833 +

9839 +

9851 +

9857 +

9887 +

9923 +

9929 +

9931 -

9941 +

10007 +

10037 +

10039 -

10061 +

10067 +

10079 +

10091 +

10133 +

10139 +

10151 +

10163 +

10169 +

10181 +

10193 +

10211 +

10223 +

10247 +

10253 +

10259 +

10271 +

10289 +

10301 +

10313 +

10331 +

10337 +

10343 +

10369 -

10391 +

10427 +

10433 +

10457 +

10463 +

10487 +

10499 +

10529 +

10589 +

10601 +

10607 +

10613 +

10631 +

10667 +

10691 +

10709 +

10733 +

10739 +

10781 +

10799 +

10847 +

10853 +

10859 +

10883 +

10889 +

10937 +

10949 +

10973 +

10979 +

10993 -

11003 +

11027 +

11057 +

11069 +

11087 +

11093 +

11117 +

11159 +

11171 +

11213 +

11243 +

11261 +

11273 +

11279 +

11317 -

11321 +

11351 +

11369 +

11393 +

11399 +

11411 +

11423 +

11443 -

11447 +

11471 +

11483 +

11489 +

11503 -

11519 +

11549 +

11579 +

11597 +

11621 +

11633 +

11657 +

11681 +

11699 +

11777 +

11783 +

11789 +

11801 +

11807 +

11813 +

11867 +

11887 -

11897 +

11903 +

11909 +

11927 +

11933 +

11939 +

11969 +

11981 +

11987 +

12011 +

12041 +

12043 -

12049 -

12071 +

12101 +

12107 +

12113 +

12119 +

12143 +

12149 +

12161 +

12197 +

12203 +

12227 +

12239 +

12251 +

12263 +

12269 +

12281 +

12301 -

12323 +

12347 +

12377 +

12401 +

12413 +

12437 +

12473 +

12479 +

12491 +

12497 +

12503 +

12527 +

12539 +

12569 +

12611 +

12641 +

12647 +

12653 +

12659 +

12671 +

12689 +

12713 +

12743 +

12763 -

12791 +

12809 +

12893 +

12899 +

12911 +

12917 +

12923 +

12941 +

12953 +

12959 +

12967 -

13001 +

13007 +

13037 +

13043 +

13049 +

13103 +

13109 +

13121 +

13127 +

13163 +

Discrepancy 3 multiplicative sequence of length 13186

2010-02-06T03:33:21Z

Thomas: page created

Here is an example of a multiplicative sequence with discrepancy 3 and length 13186. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [[http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884] this comment].

===The sequence===

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+--++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-+++-+--+-++--+--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
--++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--++++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--++++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-+--++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-+---+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+-++-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
++++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
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===Primes sent to -1===

The full list of primes sent to -1 is:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

===Sign flips===

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at:
709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967.
So the backtracking has not retained all the initial sign flips. There are many further flips, coming from primes equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) and have in fact been sent to +1.

Here are the primes where a sign flip from <math\mu_3</math> occurs, with the new sign:

prime sign after flip
2 +
5 +
11 +
17 +
23 +
29 +
41 +
47 +
53 +
59 +
71 +
83 +
89 +
101 +
107 +
113 +
131 +
137 +
149 +
167 +
173 +
179 +
191 +
197 +
227 +
233 +
239 +
251 +
257 +
263 +
269 +
281 +
293 +
311 +
317 +
347 +
353 +
359 +
383 +
389 +
401 +
419 +
431 +
443 +
449 +
461 +
467 +
479 +
491 +
503 +
509 +
521 +
557 +
563 +
569 +
587 +
593 +
599 +
617 +
641 +
647 +
653 +
659 +
677 +
683 +
701 +
709 -
719 +
743 +
761 +
773 +
797 +
809 +
821 +
827 +
839 +
857 +
863 +
887 +
911 +
929 +
941 +
947 +
953 +
971 +
977 +
983 +
1013 +
1019 +
1031 +
1049 +
1061 +
1091 +
1097 +
1103 +
1109 +
1151 +
1163 +
1181 +
1187 +
1193 +
1201 -
1217 +
1223 +
1229 +
1259 +
1277 +
1283 +
1289 +
1301 +
1307 +
1319 +
1361 +
1367 +
1373 +
1409 +
1427 +
1433 +
1439 +
1451 +
1481 +
1487 +
1493 +
1499 +
1511 +
1523 +
1553 +
1559 +
1571 +
1583 +
1601 +
1607 +
1613 +
1619 +
1637 +
1667 +
1709 +
1721 +
1733 +
1787 +
1811 +
1823 +
1847 +
1871 +
1877 +
1889 +
1901 +
1907 +
1913 +
1931 +
1949 +
1973 +
1979 +
1997 +
1999 -
2003 +
2027 +
2039 +
2063 +
2069 +
2081 +
2087 +
2099 +
2111 +
2129 +
2141 +
2153 +
2207 +
2213 +
2237 +
2243 +
2267 +
2273 +
2297 +
2309 +
2333 +
2339 +
2351 +
2357 +
2381 +
2393 +
2399 +
2411 +
2417 +
2423 +
2441 +
2447 +
2459 +
2477 +
2531 +
2543 +
2549 +
2579 +
2591 +
2609 +
2621 +
2633 +
2657 +
2663 +
2687 +
2693 +
2699 +
2711 +
2729 +
2741 +
2753 +
2777 +
2789 +
2801 +
2819 +
2843 +
2861 +
2879 +
2897 +
2903 +
2909 +
2927 +
2939 +
2957 +
2963 +
2969 +
2999 +
3011 +
3023 +
3041 +
3083 +
3089 +
3119 +
3137 +
3167 +
3191 +
3203 +
3209 +
3221 +
3251 +
3257 +
3299 +
3329 +
3347 +
3359 +
3371 +
3389 +
3407 +
3413 +
3449 +
3457 -
3461 +
3467 +
3491 +
3527 +
3533 +
3539 +
3541 -
3557 +
3581 +
3593 +
3613 -
3617 +
3623 +
3659 +
3671 +
3677 +
3701 +
3719 +
3761 +
3767 +
3779 +
3797 +
3803 +
3821 +
3833 +
3851 +
3863 +
3881 +
3911 +
3923 +
3929 +
3947 +
3989 +
4001 +
4003 -
4007 +
4013 +
4019 +
4049 +
4073 +
4079 +
4091 +
4127 +
4133 +
4139 +
4157 +
4211 +
4217 +
4219 -
4229 +
4241 +
4253 +
4271 +
4283 +
4289 +
4337 +
4349 +
4373 +
4391 +
4409 +
4421 +
4451 +
4457 +
4463 +
4481 +
4493 +
4517 +
4523 +
4547 +
4583 +
4637 +
4643 +
4649 +
4673 +
4679 +
4691 +
4703 +
4721 +
4733 +
4751 +
4787 +
4793 +
4817 +
4871 +
4877 +
4889 +
4919 +
4931 +
4937 +
4943 +
4967 +
4973 +
5003 +
5009 +
5021 +
5039 +
5081 +
5087 +
5099 +
5147 +
5153 +
5171 +
5189 +
5231 +
5237 +
5261 +
5273 +
5279 +
5297 +
5303 +
5309 +
5333 +
5351 +
5381 +
5387 +
5393 +
5399 +
5417 +
5441 +
5471 +
5477 +
5483 +
5501 +
5507 +
5519 +
5531 +
5573 +
5591 +
5639 +
5651 +
5657 +
5659 -
5669 +
5693 +
5711 +
5717 +
5741 +
5783 +
5801 +
5807 +
5813 +
5843 +
5849 +
5867 +
5879 +
5897 +
5903 +
5927 +
5939 +
5953 -
5981 +
5987 +
6007 -
6011 +
6029 +
6047 +
6053 +
6089 +
6101 +
6113 +
6131 +
6143 +
6163 -
6173 +
6197 +
6203 +
6221 +
6257 +
6263 +
6269 +
6287 +
6299 +
6311 +
6317 +
6323 +
6329 +
6353 +
6359 +
6389 +
6449 +
6473 +
6521 +
6551 +
6563 +
6569 +
6581 +
6599 +
6653 +
6659 +
6689 +
6701 +
6719 +
6737 +
6761 +
6779 +
6791 +
6803 +
6827 +
6833 +
6857 +
6863 +
6869 +
6899 +
6907 -
6911 +
6917 +
6947 +
6959 +
6971 +
6977 +
6983 +
7001 +
7013 +
7019 +
7043 +
7079 +
7103 +
7109 +
7121 +
7127 +
7151 +
7187 +
7193 +
7229 +
7247 +
7253 +
7283 +
7297 -
7307 +
7331 +
7349 +
7433 +
7451 +
7457 +
7481 +
7487 +
7499 +
7517 +
7523 +
7529 +
7541 +
7547 +
7559 +
7577 +
7583 +
7589 +
7607 +
7643 +
7649 +
7673 +
7691 +
7703 +
7727 +
7757 +
7793 +
7817 +
7823 +
7829 +
7841 +
7853 +
7877 +
7883 +
7901 +
7907 +
7919 +
7927 -
7937 +
7949 +
8009 +
8039 +
8069 +
8081 +
8087 +
8093 +
8111 +
8117 +
8123 +
8147 +
8171 +
8219 +
8231 +
8237 +
8243 +
8273 +
8297 +
8363 +
8369 +
8387 +
8423 +
8429 +
8447 +
8501 +
8537 +
8543 +
8573 +
8597 +
8609 +
8627 +
8663 +
8669 +
8681 +
8693 +
8699 +
8741 +
8747 +
8803 -
8807 +
8819 +
8831 +
8837 +
8849 +
8861 +
8867 +
8933 +
8951 +
8963 +
8969 +
8999 +
9011 +
9029 +
9041 +
9059 +
9137 +
9161 +
9173 +
9203 +
9209 +
9221 +
9239 +
9257 +
9281 +
9293 +
9311 +
9323 +
9341 +
9371 +
9377 +
9413 +
9419 +
9431 +
9437 +
9461 +
9467 +
9473 +
9479 +
9491 +
9497 +
9521 +
9533 +
9539 +
9547 -
9551 +
9619 -
9623 +
9629 +
9677 +
9719 +
9743 +
9767 +
9791 +
9803 +
9833 +
9839 +
9851 +
9857 +
9887 +
9923 +
9929 +
9931 -
9941 +
10007 +
10037 +
10039 -
10061 +
10067 +
10079 +
10091 +
10133 +
10139 +
10151 +
10163 +
10169 +
10181 +
10193 +
10211 +
10223 +
10247 +
10253 +
10259 +
10271 +
10289 +
10301 +
10313 +
10331 +
10337 +
10343 +
10369 -
10391 +
10427 +
10433 +
10457 +
10463 +
10487 +
10499 +
10529 +
10589 +
10601 +
10607 +
10613 +
10631 +
10667 +
10691 +
10709 +
10733 +
10739 +
10781 +
10799 +
10847 +
10853 +
10859 +
10883 +
10889 +
10937 +
10949 +
10973 +
10979 +
10993 -
11003 +
11027 +
11057 +
11069 +
11087 +
11093 +
11117 +
11159 +
11171 +
11213 +
11243 +
11261 +
11273 +
11279 +
11317 -
11321 +
11351 +
11369 +
11393 +
11399 +
11411 +
11423 +
11443 -
11447 +
11471 +
11483 +
11489 +
11503 -
11519 +
11549 +
11579 +
11597 +
11621 +
11633 +
11657 +
11681 +
11699 +
11777 +
11783 +
11789 +
11801 +
11807 +
11813 +
11867 +
11887 -
11897 +
11903 +
11909 +
11927 +
11933 +
11939 +
11969 +
11981 +
11987 +
12011 +
12041 +
12043 -
12049 -
12071 +
12101 +
12107 +
12113 +
12119 +
12143 +
12149 +
12161 +
12197 +
12203 +
12227 +
12239 +
12251 +
12263 +
12269 +
12281 +
12301 -
12323 +
12347 +
12377 +
12401 +
12413 +
12437 +
12473 +
12479 +
12491 +
12497 +
12503 +
12527 +
12539 +
12569 +
12611 +
12641 +
12647 +
12653 +
12659 +
12671 +
12689 +
12713 +
12743 +
12763 -
12791 +
12809 +
12893 +
12899 +
12911 +
12917 +
12923 +
12941 +
12953 +
12959 +
12967 -
13001 +
13007 +
13037 +
13043 +
13049 +
13103 +
13109 +
13121 +
13127 +
13163 +

Multiplicative sequences

2010-02-06T03:24:12Z

Thomas: created separated page for lenght 13186 example

==Case C=2==

Any completely-multiplicative sequence of length <math>247</math> has discrepancy more than <math>2</math>.

===Data and plots===

There are 500 sequences of length <math>246</math> with discrepancy <math>2</math>, all of which agree at primes up to and including <math>67</math>. Here is one example:

0 + - - + - + - - + + + - - + + + - -
+ - + - - + + + - - + - + - - + + + -
- + + - - - + - + - - + - + - + + - +
- - + + + - - + + + - - + - - - + + -
+ - - + - + + - + + + - - - + + - - +
- + - - + + - - + + - - + - + + + - -
+ + + - - + - + - + + - + - - + - + -
- + + + - - - + + + - + - - - - + + +
- - + - + + - - + + - - - + + - - + -
+ - + + - + - + + - + - - + + + - - +
+ - - - + - + - - + - + + - + + - - -
+ + - + + - + + - - - - + - + + + + -
- - - + - + + + + - - - + + - - + - -

Here are the values this sequence takes at the first few primes. The primes up to 101 that go to -1 are 2, 3, 5, 7, 13, 17, 23, 37, 41, 43, 47, 67, 71, 83, 89, 97, 101. The primes less than 100 that go to 1 are 11, 19, 29, 31, 53, 59, 61, 73, 79.

The total number of such multiplicative sequences for each length can be generated with Alec's python script.

Here is a [http://thomas1111.files.wordpress.com/2010/01/multnumzoom.png plot of this data], and [http://www.obtext.com/erdos/log_number_of_cm_seqs.png a plot of the log of the data], and here are the precise numbers:

length number

2 2

3 3

4 3

5 4

6 4

7 7

8 7

9 6

10 6

11 10

12 10

13 15

14 15

15 14

16 14

17 21

18 21

19 34

20 34

21 24

22 24

23 38

24 38

25 28

26 28

27 23

28 23

29 34

30 34

31 54

32 54

33 37

34 37

35 28

36 28

37 40

38 40

39 31

40 31

41 48

42 48

43 72

44 72

45 57

46 57

47 89

48 89

49 81

50 81

51 62

52 62

53 92

54 92

55 55

56 55

57 44

58 44

59 68

60 68

61 111

62 111

63 83

64 83

65 71

66 71

67 113

68 113

69 97

70 97

71 157

72 157

73 240

74 240

75 175

76 175

77 125

78 125

79 185

80 185

81 178

82 178

83 286

84 286

85 212

86 212

87 178

88 178

89 276

90 276

91 163

92 163

93 138

94 138

95 119

96 119

97 176

98 176

99 129

100 129

101 198

102 198

103 315

104 315

105 277

106 277

107 426

108 426

109 656

110 656

111 485

112 485

113 846

114 846

115 502

116 502

117 256

118 256

119 198

120 198

121 112

122 112

123 82

124 82

125 82

126 82

127 100

128 100

129 84

130 84

131 134

132 134

133 56

134 56

135 44

136 44

137 61

138 61

139 105

140 105

141 84

142 84

143 72

144 72

145 55

146 55

147 48

148 48

149 72

150 72

151 120

152 120

153 72

154 72

155 72

156 72

157 132

158 132

159 112

160 112

161 112

162 112

163 184

164 184

165 164

166 164

167 246

168 246

169 234

170 234

171 168

172 168

173 246

174 246

175 246

176 246

177 246

178 246

179 408

180 408

181 624

182 624

183 414

184 414

185 384

186 384

187 286

188 286

189 286

190 286

191 304

192 304

193 392

194 392

195 362

196 362

197 468

198 468

199 812

200 812

201 776

202 776

203 626

204 626

205 386

206 386

207 386

208 386

209 386

210 386

211 694

212 694

213 573

214 573

215 471

216 471

217 279

218 279

219 259

220 259

221 259

222 259

223 354

224 354

225 125

226 125

227 125

228 125

229 250

230 250

231 250

232 250

233 375

234 375

235 250

236 250

237 250

238 250

239 500

240 500

241 750

242 750

243 500

244 500

245 500

246 500<math>Insert formula here</math>

247 0

And the same data in Mathematica format: {{1,1},{2,2},{3,3},{4,3},{5,4},{6,4},{7,7},{8,7},{9,6},{10,6},{11,10},{12,10},{13,15},{14,15},{15,14},{16,14},{17,21},{18,21},{19,34},{20,34},{21,24},{22,24},{23,38},{24,38},{25,28},{26,28},{27,23},{28,23},{29,34},{30,34},{31,54},{32,54},{33,37},{34,37},{35,28},{36,28},{37,40},{38,40},{39,31},{40,31},{41,48},{42,48},{43,72},{44,72},{45,57},{46,57},{47,89},{48,89},{49,81},{50,81},{51,62},{52,62},{53,92},{54,92},{55,55},{56,55},{57,44},{58,44},{59,68},{60,68},{61,111},{62,111},{63,83},{64,83},{65,71},{66,71},{67,113},{68,113},{69,97},{70,97},{71,157},{72,157},{73,240},{74,240},{75,175},{76,175},{77,125},{78,125},{79,185},{80,185},{81,178},{82,178},{83,286},{84,286},{85,212},{86,212},{87,178},{88,178},{89,276},{90,276},{91,163},{92,163},{93,138},{94,138},{95,119},{96,119},{97,176},{98,176},{99,129},{100,129},{101,198},{102,198},{103,315},{104,315},{105,277},{106,277},{107,426},{108,426},{109,656},{110,656},{111,485},{112,485},{113,846},{114,846},{115,502},{116,502},{117,256},{118,256},{119,198},{120,198},{121,112},{122,112},{123,82},{124,82},{125,82},{126,82},{127,100},{128,100},{129,84},{130,84},{131,134},{132,134},{133,56},{134,56},{135,44},{136,44},{137,61},{138,61},{139,105},{140,105},{141,84},{142,84},{143,72},{144,72},{145,55},{146,55},{147,48},{148,48},{149,72},{150,72},{151,120},{152,120},{153,72},{154,72},{155,72},{156,72},{157,132},{158,132},{159,112},{160,112},{161,112},{162,112},{163,184},{164,184},{165,164},{166,164},{167,246},{168,246},{169,234},{170,234},{171,168},{172,168},{173,246},{174,246},{175,246},{176,246},{177,246},{178,246},{179,408},{180,408},{181,624},{182,624},{183,414},{184,414},{185,384},{186,384},{187,286},{188,286},{189,286},{190,286},{191,304},{192,304},{193,392},{194,392},{195,362},{196,362},{197,468},{198,468},{199,812},{200,812},{201,776},{202,776},{203,626},{204,626},{205,386},{206,386},{207,386},{208,386},{209,386},{210,386},{211,694},{212,694},{213,573},{214,573},{215,471},{216,471},{217,279},{218,279},{219,259},{220,259},{221,259},{222,259},{223,354},{224,354},{225,125},{226,125},{227,125},{228,125},{229,250},{230,250},{231,250},{232,250},{233,375},{234,375},{235,250},{236,250},{237,250},{238,250},{239,500},{240,500},{241,750},{242,750},{243,500},{244,500},{245,500},{246,500},{247,0}}

==Case C=3==

The maximum length for <math>C=3</math> is at least <math>13186</math>.

An example of that length is detailed [[Discrepancy 3 multiplicative sequence of length 13186|on this page]].

Here is an example of that length. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [[http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884] this comment].

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+-++--+--++++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++++
+--+-+--++--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+--+-++--
+-++--+-+--+++-+--+-++-++--+-++--+--+-++-+---+-+--+++-+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
---+--+-++--+--+-++-++--+++++-+----++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+-++-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++--+--+-++-++--+-++-+---+-++-++--+--+-+
+--+-++--+--+-++--+--+-++-++--+-++--+--++++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
+--+-+--++--+-++--+---+++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at: 709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967. So the backtracking has not retained all the initial sign flips, and there may be further ones by checking whether any prime equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) has in fact been sent to +1.

The full list of primes sent to -1 is: 2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

Length 1530:

+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-+--++
+--+++--+-+--+-+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-
+--+-+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-
+--+++--+++--+-++-+-+--+-+--+++--+++--+-+--+++--+-+--+++--++
+--+-+--++---+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-
+--+-+--+++--+++--+-+--+-+--+-+--+++--+++--+-+--+++--+-+--++
+--+++--+-+--+++--+-+--+++--+++----+--+++--+-+--+++--+++--+-
+--+-+--+-+--+++--+++--+-+--+-+--+-++-+++--+++--+-+--+++--+-
+--+++--+++--+-+--+-++-+-+---++--+++--+-+--+++--+-+--+++--++
+--+-+--+-+--+-+--+++--+++--+-+--+-+--+-+--+++--++--++-+--++
+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+++--+-+--++
---+++--+-+--+-+--+-+-++++--+++--+-+--+-+--+-+--+++--+++--+-
+--+++--+-+--+++--+++--+-+--+-+-++-+--++---+++----+--+++--+-
+--+++-++++--+-+--+-+--+-+--+++--+++--+-+--+-+--+-+--+++--++
+--+-+--+++--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--++
+--+-+--+++---++--+-+--+-+--+-+--+++--+++--+-+--+-+-++-+--++
+--+++--+-+--+++--+-+--+-+--+++--+-++-+++----+--+++--+++-++-
---+++--+-+--+++--+++--+-+--+-+--+-+--+++--+++--+-+--+-+--+-
+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-+--+++--++
+--+-+--+++--+-+--+++--+++--+-+--+-+--+-++-++---+++-++-+--+-
+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+++---+
+--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-+--+++--+++--+-
++-+-+--+-+--+++--+++--+-+--+++--+-+--+++--++-+-+---++-+----
+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-+--+++--++
+--+-+--+-+-++-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--++
+--+----++---+++--+-++-++++-+-

These are the primes sent to -1 in this example: 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 223, 227, 233, 251, 257, 263, 277, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 433, 443, 463, 467, 487, 503, 509, 523, 547, 557, 563, 577, 587, 607, 613, 617, 643, 647, 653, 661, 673, 677, 727, 733, 743, 757, 761, 769, 773, 787, 797, 823, 827, 853, 857, 863, 877, 883, 887, 907, 937, 947, 967, 977, 983, 1013, 1021, 1033, 1063, 1087, 1093, 1097, 1103, 1117, 1123, 1153, 1163, 1187, 1213, 1217, 1223, 1237, 1259, 1277, 1283, 1297, 1303, 1307, 1327, 1423, 1427, 1433, 1447, 1483, 1487, 1493, 1511, 1523

This came from a search initialized by sending primes congruent to 2 or 3 mod 5, and the prime 5, to -1, and others to +1.

Length 852:

+--+++--+-+--+-+--+++--+++--+++--+-+--+-+--+++--+-+--+++--+-
+--+-+--+++--+-+--+++--+-+--+-+--+++--+++--+++--+-+--+-+--++
+--+++--+++--+-+--+-+--+++--++---+++--+-+--+-+--+++--+-+--++
+--+-+--+-+--++++-+-+--+++--+-+--+-+--+++--+++--+++--+-+--+-
+--+++--+-+--+++--+-+--+-+--+++--+++--++---+-+--+-+--+++--+-
++-+++--+-+--+-+--+++--+-+--+++--+-+--+-+--+++--+++--+++--+-
+--+-+--+++--+-+--+++-++-+--+-+---++--+++--+++--+-+--+-+--++
+--+-+--+++--+-+--+-+--+++--+-+-++++--+-+--+-+--+++--+++--++
+--+-+--+-+--+++---++--+++--+-+--+-+--+++--+++--+++--+-+--+-
+--+++--+-+--+++--+-++-+-+--+++--+-+--+++--+-+--+----+++--++
+---++--+-+--+-++-+++--+++--+++--+-+--+----+++--+++--+++--+-
+--+-+--+++--+-+--+++-++-+--+-+--+++--+-+--+++--+-+--+-+--++
+--+++--+++--+-+--+-+--+++--+++--+-++-+-+--+-----++--+++--++
+--+-+-++-+--+++--+-+--+++--+-+--+-+--+++--+-+--+++--+-+--+-
+-++++--+++-

These are the primes sent to -1 in this example: 2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 151, 157, 163, 167, 173, 193, 223, 227, 233, 257, 263, 277, 281, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 397, 433, 443, 457, 463, 467, 487, 499, 503, 523, 547, 557, 563, 577, 587, 593, 607, 613, 641, 643, 647, 653, 673, 677, 727, 733, 743, 769, 773, 787, 797, 823, 827.

This came from a search initialized by sending primes congruent to 2 or 3 mod 5 to -1 and others to +1.

Note that the only primes not congruent to 2 or 3 that are sent to -1 are 151, 281, 499, 641 and 769. [Are there some that are congruent to 2 or 3 that are sent to 1? If so, which are they?]

Length 819:

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++-----+-++-++--+-+++++--+-++--+--+-++--+--+-++-++--
+-++-----+-++-++--+-++-+++-+-+---+--+-++--+--+-++-+++-+-+-+-
+--+-++--++-+-++-+---+-++--+--+-++-++-++-+--+++---++--+--+-+
+--+--+-++-+++-+-++--+--+--++----+-++-+++-+-++--+----++-++--
++++-++--+-++--+--+-++-++--+-+--++--+-++--+---+++--+--+++--+
+--+++---+--+-++-+--+-+++-++--+-++---+-+-++--+--+-++-++--+++
---+--+-+++-+--+--+-+++--+++--+-+--+-++++---++-++--+-++--+--
+-++-+---+-++-++--+-+++-+---++---+--+--++++--+-++--+-++----+
++-+-++-++--+-++--+----++-++--+--+-++--+-+++-+--+-++--+-++-+
+-+---+-++--+--+++--+++-+-++-+---++++--++---++--+--+-++--++-
-+++--+-++-+---+--+--++++-++-+-+----+-+++++--+-+--+---+++++-
+--+-++-+----+++-++--+-++--+--+-+++--+-+--+++---+-++--+--+-+
+-++----++-+++---++-+++-+--+-+--++-++-+

The primes that go to -1 in this example are:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 73, 83, 101, 107, 113, 127, 131, 137, 149, 151, 167, 197, 199, 223, 229, 233, 239, 251, 257, 263, 271, 293, 311, 317, 331, 353, 359, 367, 379, 389, 397, 401, 421, 449, 457, 463, 467, 479, 487, 491, 557, 563, 569, 587, 593, 599, 619, 631, 643, 647, 653, 661, 673, 677, 691, 709, 733, 743, 757, 761, 773, 787, 797, 809, 811

Length 627:

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++-----+-++-++--+-+++++--+-++--+--+-++--+--+-++-++--
+-++-----+-++-++--+-++-+++-+-+---+--+-++--+--+-++-+++-+-+-+-
+--+-++--++-+-++-+---+-++--+--+-++-++-++-+--+++---++--+--+-+
+--+--+-++-+++-+-++--+--+--++----+-++-+++-+-++--+----++-++--
++++-++--+-+---+--+-++-++--+-++-++--+-++--+---+++--+--+++--+
+--+++---+--+-++-+--+-+++-++--+-++---+-+-++--+--+-++-++--+++
---+--+-+++-+--+--+-+++--+++--+-+--+-++++---++-++--+-++--+--
+-++-+---+-++-++--+-+++-+---++---+--+--++++--+-++--+-++----+
++-+-++-++--+-++--+----++-++--+--+-++--+-+++-+--+-++--+-++-+
+-+---+-++--+--+++--+++-+++

A sequence of length 545 that agrees with the maximal discrepancy-2 sequences at primes up to and including 67:

+--+-+--+++--+++--+-+--+++--+-+--+++--++---+-+--+-+-++-+--++
+--+++--+-+--+-+--+-++-+++---++--+-++-++--++--+--++--+++--+-
+-++-+--+-+--+++---++--+-+--+++--+-++--++-+-++--+-+-++-+-+--
+--+++--++--+--++---++-++---++-+--++-++-+--+++--+-+--+++--++
+--+----+++--+-+--+++--+-++-+----+++-++----++++-++--++-+--+-
+-++-++--++---++-++----+--+++-+--+++--+++--+--+-+++--+---+-+
+--++++-----++++--+-++-++--+-++----+-++++----+--+-++-++++-+-
-+---+-++-+-+++----++--+++--+----+-++-+++--++++-+----++++-+-
+--+-++--+-+-+-+--+-+--+++--+++--+-+----+--+++--+++-+--++-++
-+-++

This sequence is -1 at the following primes:

2, 3, 5, 7, 13, 17, 23, 37, 41, 43, 47, 67, 73, 83, 89, 101, 109, 113, 127, 137, 139, 167, 179, 191, 199, 211, 223, 227, 233, 257, 263, 271, 277, 281, 283, 313, 317, 337, 353, 359, 383, 389, 397, 421, 439, 443, 463, 491, 503, 523, 541

==General Case==

Here is some data on how the discrepancy of completely-multiplicative sequences grows as a function of length, depending on how missing values at prime indices are chosen.

Without loss of generality, one may always set <math>x_1=+1</math>

===Uniform choice===

Here is some data when the values of <math>x_n</math> are: a given +/-1 sequence when <math>n>1</math> is one of the first <math>N</math> primes, only the value <math>-1</math> for <math>n</math> any other prime, and a multiplicatively computed value when <math>n</math> is composite.

* N=1
** <math>x_1=+1</math>, <math>x_2=+1</math>: D(100)=21, D(1000)=107, D(10000)=407
** ...

* N=2
** <math>x_1=+1</math>, <math>x_2=+1</math>, <math>x_3=+1</math>: D(100)=34, D(1000)=262, D(10000)=1190
** ...

* N=3
** <math>x_1=+1</math>, <math>x_2=+1</math>, <math>x_3=+1</math>, <math>x_5=+1</math>: D(100)=25, D(1000)=413, D(10000)=2332
** ...

===Minimizing the discrepancy D up to the next prime===

[http://thomas1111.files.wordpress.com/2010/01/gensum_3000_sh0_disc.png Here is a plot] of D(n), the discrepancy as a function of length, as well as the partial sums of the first few HAP when one starts with <math>x_1=1</math> and ask that the value at prime <math>p</math> be either +1 or -1 depending on which allows to minimize <math>D(q)</math>, where <math>q</math> is the next prime.

[http://thomas1111.files.wordpress.com/2010/01/gensum_5000_s1124a_disc.png Here is a plot] doing the same thing but instead starting with [[The first 1124-sequence|the first 1124 sequence]] as a seed.

The two plots show that the partial sums do grow at least logarithmically.

===Minimizing the sum of partial HAP sums===

A method to choose a value at an undertermined prime <math>p</math> is to choose to impose <math>x_p=+1</math> or <math>x_p=-1</math> depending on which gave the smallest quantity <math>\ell_s(q)</math>, where <math>q</math> is the next prime and <math>\ell_s(q):=\sum_{d=1}^q s_d(q)</math> with <math>s_d(q)</math> itself the partial sum of the d-HAP up to <math>q</math>.

Here is [http://thomas1111.files.wordpress.com/2010/01/gensum_3000_sh0_plain.png a plot] obtained when setting only <math>x_1=+1</math>. On it is shown the function <math>f(x):=\log (x)</math> (the very flat curve), the partial sums of the sequence and its first few HAPs, and both <math>D(n)</math> and <math>-D(n)</math>.

Multiplicative sequences

2010-02-05T20:18:31Z

Thomas: added information on the 13186 sequence

==Case C=2==

Any completely-multiplicative sequence of length <math>247</math> has discrepancy more than <math>2</math>.

===Data and plots===

There are 500 sequences of length <math>246</math> with discrepancy <math>2</math>, all of which agree at primes up to and including <math>67</math>. Here is one example:

0 + - - + - + - - + + + - - + + + - -
+ - + - - + + + - - + - + - - + + + -
- + + - - - + - + - - + - + - + + - +
- - + + + - - + + + - - + - - - + + -
+ - - + - + + - + + + - - - + + - - +
- + - - + + - - + + - - + - + + + - -
+ + + - - + - + - + + - + - - + - + -
- + + + - - - + + + - + - - - - + + +
- - + - + + - - + + - - - + + - - + -
+ - + + - + - + + - + - - + + + - - +
+ - - - + - + - - + - + + - + + - - -
+ + - + + - + + - - - - + - + + + + -
- - - + - + + + + - - - + + - - + - -

Here are the values this sequence takes at the first few primes. The primes up to 101 that go to -1 are 2, 3, 5, 7, 13, 17, 23, 37, 41, 43, 47, 67, 71, 83, 89, 97, 101. The primes less than 100 that go to 1 are 11, 19, 29, 31, 53, 59, 61, 73, 79.

The total number of such multiplicative sequences for each length can be generated with Alec's python script.

Here is a [http://thomas1111.files.wordpress.com/2010/01/multnumzoom.png plot of this data], and [http://www.obtext.com/erdos/log_number_of_cm_seqs.png a plot of the log of the data], and here are the precise numbers:

length number

2 2

3 3

4 3

5 4

6 4

7 7

8 7

9 6

10 6

11 10

12 10

13 15

14 15

15 14

16 14

17 21

18 21

19 34

20 34

21 24

22 24

23 38

24 38

25 28

26 28

27 23

28 23

29 34

30 34

31 54

32 54

33 37

34 37

35 28

36 28

37 40

38 40

39 31

40 31

41 48

42 48

43 72

44 72

45 57

46 57

47 89

48 89

49 81

50 81

51 62

52 62

53 92

54 92

55 55

56 55

57 44

58 44

59 68

60 68

61 111

62 111

63 83

64 83

65 71

66 71

67 113

68 113

69 97

70 97

71 157

72 157

73 240

74 240

75 175

76 175

77 125

78 125

79 185

80 185

81 178

82 178

83 286

84 286

85 212

86 212

87 178

88 178

89 276

90 276

91 163

92 163

93 138

94 138

95 119

96 119

97 176

98 176

99 129

100 129

101 198

102 198

103 315

104 315

105 277

106 277

107 426

108 426

109 656

110 656

111 485

112 485

113 846

114 846

115 502

116 502

117 256

118 256

119 198

120 198

121 112

122 112

123 82

124 82

125 82

126 82

127 100

128 100

129 84

130 84

131 134

132 134

133 56

134 56

135 44

136 44

137 61

138 61

139 105

140 105

141 84

142 84

143 72

144 72

145 55

146 55

147 48

148 48

149 72

150 72

151 120

152 120

153 72

154 72

155 72

156 72

157 132

158 132

159 112

160 112

161 112

162 112

163 184

164 184

165 164

166 164

167 246

168 246

169 234

170 234

171 168

172 168

173 246

174 246

175 246

176 246

177 246

178 246

179 408

180 408

181 624

182 624

183 414

184 414

185 384

186 384

187 286

188 286

189 286

190 286

191 304

192 304

193 392

194 392

195 362

196 362

197 468

198 468

199 812

200 812

201 776

202 776

203 626

204 626

205 386

206 386

207 386

208 386

209 386

210 386

211 694

212 694

213 573

214 573

215 471

216 471

217 279

218 279

219 259

220 259

221 259

222 259

223 354

224 354

225 125

226 125

227 125

228 125

229 250

230 250

231 250

232 250

233 375

234 375

235 250

236 250

237 250

238 250

239 500

240 500

241 750

242 750

243 500

244 500

245 500

246 500<math>Insert formula here</math>

247 0

And the same data in Mathematica format: {{1,1},{2,2},{3,3},{4,3},{5,4},{6,4},{7,7},{8,7},{9,6},{10,6},{11,10},{12,10},{13,15},{14,15},{15,14},{16,14},{17,21},{18,21},{19,34},{20,34},{21,24},{22,24},{23,38},{24,38},{25,28},{26,28},{27,23},{28,23},{29,34},{30,34},{31,54},{32,54},{33,37},{34,37},{35,28},{36,28},{37,40},{38,40},{39,31},{40,31},{41,48},{42,48},{43,72},{44,72},{45,57},{46,57},{47,89},{48,89},{49,81},{50,81},{51,62},{52,62},{53,92},{54,92},{55,55},{56,55},{57,44},{58,44},{59,68},{60,68},{61,111},{62,111},{63,83},{64,83},{65,71},{66,71},{67,113},{68,113},{69,97},{70,97},{71,157},{72,157},{73,240},{74,240},{75,175},{76,175},{77,125},{78,125},{79,185},{80,185},{81,178},{82,178},{83,286},{84,286},{85,212},{86,212},{87,178},{88,178},{89,276},{90,276},{91,163},{92,163},{93,138},{94,138},{95,119},{96,119},{97,176},{98,176},{99,129},{100,129},{101,198},{102,198},{103,315},{104,315},{105,277},{106,277},{107,426},{108,426},{109,656},{110,656},{111,485},{112,485},{113,846},{114,846},{115,502},{116,502},{117,256},{118,256},{119,198},{120,198},{121,112},{122,112},{123,82},{124,82},{125,82},{126,82},{127,100},{128,100},{129,84},{130,84},{131,134},{132,134},{133,56},{134,56},{135,44},{136,44},{137,61},{138,61},{139,105},{140,105},{141,84},{142,84},{143,72},{144,72},{145,55},{146,55},{147,48},{148,48},{149,72},{150,72},{151,120},{152,120},{153,72},{154,72},{155,72},{156,72},{157,132},{158,132},{159,112},{160,112},{161,112},{162,112},{163,184},{164,184},{165,164},{166,164},{167,246},{168,246},{169,234},{170,234},{171,168},{172,168},{173,246},{174,246},{175,246},{176,246},{177,246},{178,246},{179,408},{180,408},{181,624},{182,624},{183,414},{184,414},{185,384},{186,384},{187,286},{188,286},{189,286},{190,286},{191,304},{192,304},{193,392},{194,392},{195,362},{196,362},{197,468},{198,468},{199,812},{200,812},{201,776},{202,776},{203,626},{204,626},{205,386},{206,386},{207,386},{208,386},{209,386},{210,386},{211,694},{212,694},{213,573},{214,573},{215,471},{216,471},{217,279},{218,279},{219,259},{220,259},{221,259},{222,259},{223,354},{224,354},{225,125},{226,125},{227,125},{228,125},{229,250},{230,250},{231,250},{232,250},{233,375},{234,375},{235,250},{236,250},{237,250},{238,250},{239,500},{240,500},{241,750},{242,750},{243,500},{244,500},{245,500},{246,500},{247,0}}

==Case C=3==

The maximum length for <math>C=3</math> is at least <math>13186</math>.

Here is an example of that length. It comes from backtracking an example of length 3545 which was constructed as a slight modification of <math>\mu_3</math> by introducing sign flips at primes 709, 881, 1201, 1697, 1999, 2837, 3323, 3457. See [[http://gowers.wordpress.com/2010/02/05/edp6-what-are-the-chances-of-success/#comment-5884] this comment].

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+-++--+--+-++-++--+-++--+--+++--++--+-++-++--+-++--+--+-++--
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+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--+--+-++-++--+-+++-+----++-++--+-++-++--+-++--+--+-+
+-++--+--+-++--+-++--+--+-++--+--++++-++--+-++--+--+-++-++--
+-++-++--+-++--+----++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-++-++---+++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++-++-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-+
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++----++-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--++-
+-++-++--+--+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+-++-++--+--+--+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--
+--+-++--+--+-++--+--+-++-++--+-++-++-++-++--+--+-++--+--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++----++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++---+++-++--+-++--+--+-++--
+----+++++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+----++--+--++++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++-+---+-++-++--+-++--
+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-+
+--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-+
+--+-++--+--+-+++++----++-++--+-++--+--+-++-++--+--+-++--+-+
+--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--++++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++++
+--+-+--++--+-++--+--+-+++-+--+-++-++--+-++--+--+-++--+--+-+
+-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+--+-++--
+-++--+-+--+++-+--+-++-++--+-++--+--+-++-+---+-+--+++-+-++--
+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-+
+-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--
+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-+
+--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-+
---+--+-++--+--+-++-++--+++++-+----++--+--+-++-++--+-++--+--
+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--
+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--
+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+-++-+
+--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-++--
+-++-++--+-++--+--+-++--+--+-++-++--+-++-+---+-++-++--+--+-+
+--+-++--+--+-++--+--+-++-++--+-++--+--++++--+--+-++-++--+-+
+--+--+-++-++--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--
+-++--+--+-++-++--+-++--+--+-++-++--+-++-++--+-++--+--+-++-+
+--+-+--++--+-++--+---+++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++--+--+-++-++--
+-++--+--+-++-++--+-++-++--+-++--+--+-++--+--+-++-++--+-++--
+--+-++--++-+-++-++--+-++--+--+-++-++--+-++-++

Using the first part of the definition of <math>\mu_3</math> (i.e. that primes which are equal to 1 mod 3 should be sent to +1) this sequence has sign flips at: 709, 1201, 1999, 3457, 3541, 3613, 4003, 4219, 5659, 5953, 6007, 6163, 6907, 7297, 7927, 8803, 9547, 9619, 9931, 10039, 10369, 10993, 11317, 11443, 11503, 11887, 12043, 12049, 12301, 12763, 12967. So the backtracking has not retained all the initial sign flips, and there may be further ones by checking whether any prime equal to 2 mod 3 (which <math>\mu_3</math> would send to -1) has in fact been sent to +1.

The full list of primes sent to -1 is: 2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641, 647, 653, 659, 677, 683, 701, 709, 719, 743, 761, 773, 797, 809, 821, 827, 839, 857, 863, 887, 911, 929, 941, 947, 953, 971, 977, 983, 1013, 1019, 1031, 1049, 1061, 1091, 1097, 1103, 1109, 1151, 1163, 1181, 1187, 1193, 1201, 1217, 1223, 1229, 1259, 1277, 1283, 1289, 1301, 1307, 1319, 1361, 1367, 1373, 1409, 1427, 1433, 1439, 1451, 1481, 1487, 1493, 1499, 1511, 1523, 1553, 1559, 1571, 1583, 1601, 1607, 1613, 1619, 1637, 1667, 1709, 1721, 1733, 1787, 1811, 1823, 1847, 1871, 1877, 1889, 1901, 1907, 1913, 1931, 1949, 1973, 1979, 1997, 1999, 2003, 2027, 2039, 2063, 2069, 2081, 2087, 2099, 2111, 2129, 2141, 2153, 2207, 2213, 2237, 2243, 2267, 2273, 2297, 2309, 2333, 2339, 2351, 2357, 2381, 2393, 2399, 2411, 2417, 2423, 2441, 2447, 2459, 2477, 2531, 2543, 2549, 2579, 2591, 2609, 2621, 2633, 2657, 2663, 2687, 2693, 2699, 2711, 2729, 2741, 2753, 2777, 2789, 2801, 2819, 2843, 2861, 2879, 2897, 2903, 2909, 2927, 2939, 2957, 2963, 2969, 2999, 3011, 3023, 3041, 3083, 3089, 3119, 3137, 3167, 3191, 3203, 3209, 3221, 3251, 3257, 3299, 3329, 3347, 3359, 3371, 3389, 3407, 3413, 3449, 3457, 3461, 3467, 3491, 3527, 3533, 3539, 3541, 3557, 3581, 3593, 3613, 3617, 3623, 3659, 3671, 3677, 3701, 3719, 3761, 3767, 3779, 3797, 3803, 3821, 3833, 3851, 3863, 3881, 3911, 3923, 3929, 3947, 3989, 4001, 4003, 4007, 4013, 4019, 4049, 4073, 4079, 4091, 4127, 4133, 4139, 4157, 4211, 4217, 4219, 4229, 4241, 4253, 4271, 4283, 4289, 4337, 4349, 4373, 4391, 4409, 4421, 4451, 4457, 4463, 4481, 4493, 4517, 4523, 4547, 4583, 4637, 4643, 4649, 4673, 4679, 4691, 4703, 4721, 4733, 4751, 4787, 4793, 4817, 4871, 4877, 4889, 4919, 4931, 4937, 4943, 4967, 4973, 5003, 5009, 5021, 5039, 5081, 5087, 5099, 5147, 5153, 5171, 5189, 5231, 5237, 5261, 5273, 5279, 5297, 5303, 5309, 5333, 5351, 5381, 5387, 5393, 5399, 5417, 5441, 5471, 5477, 5483, 5501, 5507, 5519, 5531, 5573, 5591, 5639, 5651, 5657, 5659, 5669, 5693, 5711, 5717, 5741, 5783, 5801, 5807, 5813, 5843, 5849, 5867, 5879, 5897, 5903, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6047, 6053, 6089, 6101, 6113, 6131, 6143, 6163, 6173, 6197, 6203, 6221, 6257, 6263, 6269, 6287, 6299, 6311, 6317, 6323, 6329, 6353, 6359, 6389, 6449, 6473, 6521, 6551, 6563, 6569, 6581, 6599, 6653, 6659, 6689, 6701, 6719, 6737, 6761, 6779, 6791, 6803, 6827, 6833, 6857, 6863, 6869, 6899, 6907, 6911, 6917, 6947, 6959, 6971, 6977, 6983, 7001, 7013, 7019, 7043, 7079, 7103, 7109, 7121, 7127, 7151, 7187, 7193, 7229, 7247, 7253, 7283, 7297, 7307, 7331, 7349, 7433, 7451, 7457, 7481, 7487, 7499, 7517, 7523, 7529, 7541, 7547, 7559, 7577, 7583, 7589, 7607, 7643, 7649, 7673, 7691, 7703, 7727, 7757, 7793, 7817, 7823, 7829, 7841, 7853, 7877, 7883, 7901, 7907, 7919, 7927, 7937, 7949, 8009, 8039, 8069, 8081, 8087, 8093, 8111, 8117, 8123, 8147, 8171, 8219, 8231, 8237, 8243, 8273, 8297, 8363, 8369, 8387, 8423, 8429, 8447, 8501, 8537, 8543, 8573, 8597, 8609, 8627, 8663, 8669, 8681, 8693, 8699, 8741, 8747, 8803, 8807, 8819, 8831, 8837, 8849, 8861, 8867, 8933, 8951, 8963, 8969, 8999, 9011, 9029, 9041, 9059, 9137, 9161, 9173, 9203, 9209, 9221, 9239, 9257, 9281, 9293, 9311, 9323, 9341, 9371, 9377, 9413, 9419, 9431, 9437, 9461, 9467, 9473, 9479, 9491, 9497, 9521, 9533, 9539, 9547, 9551, 9619, 9623, 9629, 9677, 9719, 9743, 9767, 9791, 9803, 9833, 9839, 9851, 9857, 9887, 9923, 9929, 9931, 9941, 10007, 10037, 10039, 10061, 10067, 10079, 10091, 10133, 10139, 10151, 10163, 10169, 10181, 10193, 10211, 10223, 10247, 10253, 10259, 10271, 10289, 10301, 10313, 10331, 10337, 10343, 10369, 10391, 10427, 10433, 10457, 10463, 10487, 10499, 10529, 10589, 10601, 10607, 10613, 10631, 10667, 10691, 10709, 10733, 10739, 10781, 10799, 10847, 10853, 10859, 10883, 10889, 10937, 10949, 10973, 10979, 10993, 11003, 11027, 11057, 11069, 11087, 11093, 11117, 11159, 11171, 11213, 11243, 11261, 11273, 11279, 11317, 11321, 11351, 11369, 11393, 11399, 11411, 11423, 11443, 11447, 11471, 11483, 11489, 11503, 11519, 11549, 11579, 11597, 11621, 11633, 11657, 11681, 11699, 11777, 11783, 11789, 11801, 11807, 11813, 11867, 11887, 11897, 11903, 11909, 11927, 11933, 11939, 11969, 11981, 11987, 12011, 12041, 12043, 12049, 12071, 12101, 12107, 12113, 12119, 12143, 12149, 12161, 12197, 12203, 12227, 12239, 12251, 12263, 12269, 12281, 12301, 12323, 12347, 12377, 12401, 12413, 12437, 12473, 12479, 12491, 12497, 12503, 12527, 12539, 12569, 12611, 12641, 12647, 12653, 12659, 12671, 12689, 12713, 12743, 12763, 12791, 12809, 12893, 12899, 12911, 12917, 12923, 12941, 12953, 12959, 12967, 13001, 13007, 13037, 13043, 13049, 13103, 13109, 13121, 13127, 13163

Length 1530:

+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-+--++
+--+++--+-+--+-+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-
+--+-+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-
+--+++--+++--+-++-+-+--+-+--+++--+++--+-+--+++--+-+--+++--++
+--+-+--++---+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-
+--+-+--+++--+++--+-+--+-+--+-+--+++--+++--+-+--+++--+-+--++
+--+++--+-+--+++--+-+--+++--+++----+--+++--+-+--+++--+++--+-
+--+-+--+-+--+++--+++--+-+--+-+--+-++-+++--+++--+-+--+++--+-
+--+++--+++--+-+--+-++-+-+---++--+++--+-+--+++--+-+--+++--++
+--+-+--+-+--+-+--+++--+++--+-+--+-+--+-+--+++--++--++-+--++
+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+++--+-+--++
---+++--+-+--+-+--+-+-++++--+++--+-+--+-+--+-+--+++--+++--+-
+--+++--+-+--+++--+++--+-+--+-+-++-+--++---+++----+--+++--+-
+--+++-++++--+-+--+-+--+-+--+++--+++--+-+--+-+--+-+--+++--++
+--+-+--+++--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--++
+--+-+--+++---++--+-+--+-+--+-+--+++--+++--+-+--+-+-++-+--++
+--+++--+-+--+++--+-+--+-+--+++--+-++-+++----+--+++--+++-++-
---+++--+-+--+++--+++--+-+--+-+--+-+--+++--+++--+-+--+-+--+-
+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-+--+++--++
+--+-+--+++--+-+--+++--+++--+-+--+-+--+-++-++---+++-++-+--+-
+--+-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+++---+
+--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-+--+++--+++--+-
++-+-+--+-+--+++--+++--+-+--+++--+-+--+++--++-+-+---++-+----
+--+++--+++--+-+--+++--+-+--+++--+++--+-+--+-+--+-+--+++--++
+--+-+--+-+-++-+--+++--+++--+-+--+++--+-+--+++--+++--+-+--++
+--+----++---+++--+-++-++++-+-

These are the primes sent to -1 in this example: 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 223, 227, 233, 251, 257, 263, 277, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 433, 443, 463, 467, 487, 503, 509, 523, 547, 557, 563, 577, 587, 607, 613, 617, 643, 647, 653, 661, 673, 677, 727, 733, 743, 757, 761, 769, 773, 787, 797, 823, 827, 853, 857, 863, 877, 883, 887, 907, 937, 947, 967, 977, 983, 1013, 1021, 1033, 1063, 1087, 1093, 1097, 1103, 1117, 1123, 1153, 1163, 1187, 1213, 1217, 1223, 1237, 1259, 1277, 1283, 1297, 1303, 1307, 1327, 1423, 1427, 1433, 1447, 1483, 1487, 1493, 1511, 1523

This came from a search initialized by sending primes congruent to 2 or 3 mod 5, and the prime 5, to -1, and others to +1.

Length 852:

+--+++--+-+--+-+--+++--+++--+++--+-+--+-+--+++--+-+--+++--+-
+--+-+--+++--+-+--+++--+-+--+-+--+++--+++--+++--+-+--+-+--++
+--+++--+++--+-+--+-+--+++--++---+++--+-+--+-+--+++--+-+--++
+--+-+--+-+--++++-+-+--+++--+-+--+-+--+++--+++--+++--+-+--+-
+--+++--+-+--+++--+-+--+-+--+++--+++--++---+-+--+-+--+++--+-
++-+++--+-+--+-+--+++--+-+--+++--+-+--+-+--+++--+++--+++--+-
+--+-+--+++--+-+--+++-++-+--+-+---++--+++--+++--+-+--+-+--++
+--+-+--+++--+-+--+-+--+++--+-+-++++--+-+--+-+--+++--+++--++
+--+-+--+-+--+++---++--+++--+-+--+-+--+++--+++--+++--+-+--+-
+--+++--+-+--+++--+-++-+-+--+++--+-+--+++--+-+--+----+++--++
+---++--+-+--+-++-+++--+++--+++--+-+--+----+++--+++--+++--+-
+--+-+--+++--+-+--+++-++-+--+-+--+++--+-+--+++--+-+--+-+--++
+--+++--+++--+-+--+-+--+++--+++--+-++-+-+--+-----++--+++--++
+--+-+-++-+--+++--+-+--+++--+-+--+-+--+++--+-+--+++--+-+--+-
+-++++--+++-

These are the primes sent to -1 in this example: 2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 127, 137, 151, 157, 163, 167, 173, 193, 223, 227, 233, 257, 263, 277, 281, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 397, 433, 443, 457, 463, 467, 487, 499, 503, 523, 547, 557, 563, 577, 587, 593, 607, 613, 641, 643, 647, 653, 673, 677, 727, 733, 743, 769, 773, 787, 797, 823, 827.

This came from a search initialized by sending primes congruent to 2 or 3 mod 5 to -1 and others to +1.

Note that the only primes not congruent to 2 or 3 that are sent to -1 are 151, 281, 499, 641 and 769. [Are there some that are congruent to 2 or 3 that are sent to 1? If so, which are they?]

Length 819:

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++-----+-++-++--+-+++++--+-++--+--+-++--+--+-++-++--
+-++-----+-++-++--+-++-+++-+-+---+--+-++--+--+-++-+++-+-+-+-
+--+-++--++-+-++-+---+-++--+--+-++-++-++-+--+++---++--+--+-+
+--+--+-++-+++-+-++--+--+--++----+-++-+++-+-++--+----++-++--
++++-++--+-++--+--+-++-++--+-+--++--+-++--+---+++--+--+++--+
+--+++---+--+-++-+--+-+++-++--+-++---+-+-++--+--+-++-++--+++
---+--+-+++-+--+--+-+++--+++--+-+--+-++++---++-++--+-++--+--
+-++-+---+-++-++--+-+++-+---++---+--+--++++--+-++--+-++----+
++-+-++-++--+-++--+----++-++--+--+-++--+-+++-+--+-++--+-++-+
+-+---+-++--+--+++--+++-+-++-+---++++--++---++--+--+-++--++-
-+++--+-++-+---+--+--++++-++-+-+----+-+++++--+-+--+---+++++-
+--+-++-+----+++-++--+-++--+--+-+++--+-+--+++---+-++--+--+-+
+-++----++-+++---++-+++-+--+-+--++-++-+

The primes that go to -1 in this example are:

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 73, 83, 101, 107, 113, 127, 131, 137, 149, 151, 167, 197, 199, 223, 229, 233, 239, 251, 257, 263, 271, 293, 311, 317, 331, 353, 359, 367, 379, 389, 397, 401, 421, 449, 457, 463, 467, 479, 487, 491, 557, 563, 569, 587, 593, 599, 619, 631, 643, 647, 653, 661, 673, 677, 691, 709, 733, 743, 757, 761, 773, 787, 797, 809, 811

Length 627:

+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+
+-++--+-++-----+-++-++--+-+++++--+-++--+--+-++--+--+-++-++--
+-++-----+-++-++--+-++-+++-+-+---+--+-++--+--+-++-+++-+-+-+-
+--+-++--++-+-++-+---+-++--+--+-++-++-++-+--+++---++--+--+-+
+--+--+-++-+++-+-++--+--+--++----+-++-+++-+-++--+----++-++--
++++-++--+-+---+--+-++-++--+-++-++--+-++--+---+++--+--+++--+
+--+++---+--+-++-+--+-+++-++--+-++---+-+-++--+--+-++-++--+++
---+--+-+++-+--+--+-+++--+++--+-+--+-++++---++-++--+-++--+--
+-++-+---+-++-++--+-+++-+---++---+--+--++++--+-++--+-++----+
++-+-++-++--+-++--+----++-++--+--+-++--+-+++-+--+-++--+-++-+
+-+---+-++--+--+++--+++-+++

A sequence of length 545 that agrees with the maximal discrepancy-2 sequences at primes up to and including 67:

+--+-+--+++--+++--+-+--+++--+-+--+++--++---+-+--+-+-++-+--++
+--+++--+-+--+-+--+-++-+++---++--+-++-++--++--+--++--+++--+-
+-++-+--+-+--+++---++--+-+--+++--+-++--++-+-++--+-+-++-+-+--
+--+++--++--+--++---++-++---++-+--++-++-+--+++--+-+--+++--++
+--+----+++--+-+--+++--+-++-+----+++-++----++++-++--++-+--+-
+-++-++--++---++-++----+--+++-+--+++--+++--+--+-+++--+---+-+
+--++++-----++++--+-++-++--+-++----+-++++----+--+-++-++++-+-
-+---+-++-+-+++----++--+++--+----+-++-+++--++++-+----++++-+-
+--+-++--+-+-+-+--+-+--+++--+++--+-+----+--+++--+++-+--++-++
-+-++

This sequence is -1 at the following primes:

2, 3, 5, 7, 13, 17, 23, 37, 41, 43, 47, 67, 73, 83, 89, 101, 109, 113, 127, 137, 139, 167, 179, 191, 199, 211, 223, 227, 233, 257, 263, 271, 277, 281, 283, 313, 317, 337, 353, 359, 383, 389, 397, 421, 439, 443, 463, 491, 503, 523, 541

==General Case==

Here is some data on how the discrepancy of completely-multiplicative sequences grows as a function of length, depending on how missing values at prime indices are chosen.

Without loss of generality, one may always set <math>x_1=+1</math>

===Uniform choice===

Here is some data when the values of <math>x_n</math> are: a given +/-1 sequence when <math>n>1</math> is one of the first <math>N</math> primes, only the value <math>-1</math> for <math>n</math> any other prime, and a multiplicatively computed value when <math>n</math> is composite.

* N=1
** <math>x_1=+1</math>, <math>x_2=+1</math>: D(100)=21, D(1000)=107, D(10000)=407
** ...

* N=2
** <math>x_1=+1</math>, <math>x_2=+1</math>, <math>x_3=+1</math>: D(100)=34, D(1000)=262, D(10000)=1190
** ...

* N=3
** <math>x_1=+1</math>, <math>x_2=+1</math>, <math>x_3=+1</math>, <math>x_5=+1</math>: D(100)=25, D(1000)=413, D(10000)=2332
** ...

===Minimizing the discrepancy D up to the next prime===

[http://thomas1111.files.wordpress.com/2010/01/gensum_3000_sh0_disc.png Here is a plot] of D(n), the discrepancy as a function of length, as well as the partial sums of the first few HAP when one starts with <math>x_1=1</math> and ask that the value at prime <math>p</math> be either +1 or -1 depending on which allows to minimize <math>D(q)</math>, where <math>q</math> is the next prime.

[http://thomas1111.files.wordpress.com/2010/01/gensum_5000_s1124a_disc.png Here is a plot] doing the same thing but instead starting with [[The first 1124-sequence|the first 1124 sequence]] as a seed.

The two plots show that the partial sums do grow at least logarithmically.

===Minimizing the sum of partial HAP sums===

A method to choose a value at an undertermined prime <math>p</math> is to choose to impose <math>x_p=+1</math> or <math>x_p=-1</math> depending on which gave the smallest quantity <math>\ell_s(q)</math>, where <math>q</math> is the next prime and <math>\ell_s(q):=\sum_{d=1}^q s_d(q)</math> with <math>s_d(q)</math> itself the partial sum of the d-HAP up to <math>q</math>.

Here is [http://thomas1111.files.wordpress.com/2010/01/gensum_3000_sh0_plain.png a plot] obtained when setting only <math>x_1=+1</math>. On it is shown the function <math>f(x):=\log (x)</math> (the very flat curve), the partial sums of the sequence and its first few HAPs, and both <math>D(n)</math> and <math>-D(n)</math>.

The Erdős discrepancy problem

2010-02-05T16:46:59Z

Thomas: added definition of mu_3 and relevant link

If you want to you can [http://michaelnielsen.org/polymath1/index.php?title=Experimental_results jump straight to the main experimental results page].

==Introduction and statement of problem==

Let <math>(x_n)</math> be a <math>\scriptstyle \pm 1</math> sequence and let <math>C</math> be a constant. Must there exist positive integers <math> d,k </math> such that

:<math> \left| \sum_{i=1}^k x_{id} \right| > C </math>?

Known colloquially as "The Erdős discrepancy problem", this question has remained unanswered since the 1930s (Erdős, 1957) and Erdős offered $500 for an answer. It was also asked by Chudakov (1956). It is extremely easy to state, and a very appealing problem, so perhaps Polymath can succeed where individual mathematicians have failed. If so, then, given the notoriety of the problem, it would be a significant coup for the multiply collaborative approach to mathematics. But even if a full solution is not reached, preliminary investigations have already thrown up some interesting ideas which could perhaps be developed into publishable results.

It seems likely that the answer to Erdős's question is yes. If it is, then an easy compactness argument tells us that for every <math>C</math> there exists <math>n</math> such that for every <math>\scriptstyle \pm 1</math> sequence of length <math>n</math> there exist <math>d,k</math> with <math>dk\leq n</math> such that <math> \left| \sum_{i=1}^k x_{id} \right| > C </math>. In view of this, we make some definitions that allow one to talk about the dependence between <math>C</math> and <math>n</math>. For any finite set <math>A</math> of integers, we define the <em>error</em> on <math>A</math> by
:<math> E(A) := \sum_{a\in A} x_a </math>,
and for a set <math>\mathcal{A}</math> of finite sets of integers, we define the <em>discrepancy</em>
:<math> \delta(\mathcal{A},x) := \sup_{A\in \mathcal{A}} |E(A)|. </math>
We can think of the values taken by the sequence <math>(x_n)</math> as a red/blue colouring of the integers that tries to make the number of reds and blues in each <math>\scriptstyle A\in\mathcal{A}</math> as equal as possible. The discrepancy measures the extent to which the sequence fails in this attempt. Taking <math>\scriptstyle \mathcal{HAP}(N)</math> to be the set of <em>homogeneous arithmetic progressions</em> <math>\{d, 2d, 3d, ..., nd\}</math> contained in <math>\{1, 2, ..., N\}</math>, we can restate the question as whether <math>\scriptstyle \delta(\mathcal{HAP}(N),x) \to \infty</math> for every sequence <math>x</math>.

Two related questions have already been solved in the literature. Letting <math>\scriptstyle \mathcal{AP}(N) </math> be the collection of all (not necessarily homogeneous) arithmetic progressions in <math>\{1, 2, ..., N\}</math>, Roth (1964) proved that <math> \scriptstyle \delta(\mathcal{AP}(N),x) \geq c n^{1/4}</math>, independent of the sequence <math>x</math>. Letting <math> \scriptstyle \mathcal{HQAP}(N) </math> be the collection of all homogeneous quasi-arithmetic progressions <math>\scriptstyle\{\lfloor \alpha \rfloor,\lfloor 2\alpha \rfloor,\dots,\lfloor k\alpha \rfloor\}</math> contained in <math>\{1, 2, ..., N\}</math>, Vijay (2008) proved that <math>\scriptstyle \delta(\mathcal{HQAP}(N),x) \geq 0.02 n^{1/6}</math>, independent of the sequence <math>x</math>.

==Some definitions and notational conventions==

A <em>homogeneous arithmetic progression</em>, or HAP, is an arithmetic progression of the form <math> \{d,2d,3d,...,nd\}</math>.

When <math> x</math> is clear from context, we write <math> \delta(N) </math> in place of <math> \delta(\mathcal{HAP}(N),x) </math>.

The <em>discrepancy</em> of a sequence <math>(x_n)</math> is the supremum of <math>|\sum_{n\in P}x_n|</math> over all homogeneous arithmetic progressions P. (Strictly speaking, we should call this something like the HAP-discrepancy, but since we will almost always be talking about HAPs, we adopt the convention that "discrepancy" always means "HAP-discrepancy" unless it is stated otherwise.)
<math>Insert formula here</math>
A sequence <math>(x_n)</math> <em>has discrepancy at most</em> φ(n) if for every natural number N the maximum value of <math>|\sum_{n\in P}x_n|</math> over all homogeneous arithmetic progressions P that are subsets of {1,2,...,N} is at most φ(N).

The EDP is the Erdős discrepancy problem. (This may conceivably be changed if enough people don't like it.)

A sequence <math>(x_n)</math> is <em>completely multiplicative</em> if <math>x_{mn}=x_mx_n</math> for any two positive integers m and n. We shall sometimes abbreviate this to "multiplicative", but the reader should be aware that the word "multiplicative" normally refers to the more general class of sequences such that <math>x_{mn}=x_mx_n</math> whenever m and n are coprime. A completely multiplicative function is determined by the values it takes at primes. The <em>Liouville function</em> λ is the unique completely multiplicative function that takes the value -1 at every prime: if the prime factorization of n is <math>\prod p_i^{a_i}</math> then λ(n) equals <math>(-1)^{\sum a_i}</math>. Another important multiplicative function is <math>\mu_3</math>, the multiplicative function that’s -1 at 3, 1 at numbers that are 1 mod 3 and -1 at numbers that are 2 mod 3. This function has mean-square partial sums which grow logarithmically, [http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/#comment-5585 see this calculation].

A <em>HAP-subsequence</em> of a sequence <math>(x_n)</math> is a subsequence of the form <math>x_d,x_{2d},x_{3d},...</math> for some d. If <math>(x_n)</math> is a multiplicative <math>\pm 1</math> sequence, then every HAP-subsequence is equal to either <math>(x_n)</math> or <math>(-x_n)</math>. We shall call a sequence <em>weakly multiplicative</em> if it has only finitely many distinct HAP-subsequences. Let us also call a <math>\pm 1</math> sequence <em>quasi-multiplicative</em> if it is a composition of a completely multiplicative function from <math>\mathbb{N}</math> to an Abelian group G with a function from G to {-1,1} (This definition is too general. See [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4843 this and the next comment]). [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4705 It can be shown] that if <math>(x_n)</math> is a weakly multiplicative sequence, then it has an HAP-subsequence that is quasi-multiplicative.

It is convenient to define the maps <math>T_k</math> for <math>k \geq 1</math> by <math>T_k(x)_n = x_{kn}</math>, where <math>x = (x_1, x_2, ...)</math> is a sequence.

==Simple observations==

*To answer the problem negatively, it is sufficient to find a completely multiplicative sequence (that is, a sequence <math>(x_n)</math> such that <math>x_{mn}=x_mx_n</math> for every m,n) such that its partial sums <math>x_1+\dots+x_n</math> are bounded. First mentioned in the proof of the theorem in [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/ this] post.

*The function that takes n to 1 if the last non-zero digit of n in its ternary representation is 1 and -1 if the last non-zero digit is 2 is completely multiplicative and the partial sum up to n is easily shown to be at most <math>\log_3n +1</math>. Therefore, the rate at which the worst discrepancy grows, as a function of the length of the homogeneous progression, can be as slow as logarithmic. [[Matryoshka_Sequences|More examples of this sort are here.]] [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/] <s>[http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4553 It turns out] that the base can be made significantly higher than 3, so this example is not best possible.</s> [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4757 Correction]

*To answer the problem negatively, it is also sufficient to find a completely multiplicative function f from <math>\mathbb{N}</math> to a finite Abelian group G (meaning that f(mn)=f(m)+f(n) for every m,n) and a function h from G to {-1,1} such that for each g in G there exists d with f(d)=g and with the partial sums hf(d)+hf(2d)+...+hf(nd) bounded. In that case, one can set <math>x_n=hf(n)</math>. (Though this observation is simple with the benefit of hindsight, it might not have been made had it not been for the fact that this sort of structure was identified in a long sequence that had been produced experimentally). First mentioned [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4696 here]. Se also [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/ this post] and [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4717 this comment].

*[http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4705 It can be shown] that if a <math>\pm 1</math> sequence <math>(x_n)</math> starts with 1 and has only finitely many distinct subsequences of the form <math>(x_d,x_{2d},x_{3d},\dots)</math>, then it must have a subsequence that arises in the above way. (This was mentioned just above in the terminology section.)

*The problem for the positive integers is equivalent to the problem for the positive rationals. [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4676](Sketch proof: if you have a function f from the positive integers to {-1,1} that works, then define <math>f_q(x)</math> to be f(q!x) whenever q!x is an integer. Then take a pointwise limit of a subsequence of the functions <math>f_q</math>. [[Limits with better properties|Click here for a more detailed discussion of this construction and what it is good for]].

*Define the [[drift]] of a sequence to be the maximal value of |f(md)+...+f(nd)|. The discrepancy problem is equivalent to showing that the drift is always infinite. It is obvious that it is at least 2 (because |f(2)+f(4)|, |f(2)+f(3)|, |f(3)+f(4)| cannot all be at most 1); one can show that [[drift|it is at least 3]] (which implies as a corollary that the discrepancy is at least 2). One can also show that the [[upper and lower discrepancy]] are each at least 2.

*One can [[topological dynamics formulation|formulate the problem using topological dynamics]].

*Using Fourier analysis, one can [[Fourier reduction|reduce the problem to one about completely multiplicative functions]].

*Here is an [[algorithm for finding multiplicative sequences with bounded discrepancy]].

*Here is a page whose aim is to record a [[human proof that completely multiplicative sequences have discrepancy at least 2]].

*Here is an argument that shows that [[bounded discrepancy multiplicative functions do not correlate with characters]].

* The answer to a [[function field version]] of the problem seems to be negative.

==Experimental evidence==

''Main page: [[Experimental results]]''

A nice feature of the problem is that it lends itself well to investigation on a computer. Although the number of <math>\pm 1</math> sequences grows exponentially, the constraints on these sequences are such that depth-first searches can lead quite quickly to interesting examples. For instance, at the time of writing they have yielded several examples of sequences of length 1124 with discrepancy 2. (This is the current record.) See [[Experimental results|this page]] for more details and further links.

Another sort of evidence is to bound the discrepancy for specific sequences. For example, setting <math>x_1=-x_2=1</math> and <math> x_n=-x_{n/3}</math> or <math>x_n=x_{n-3}</math> according to whether <math>n</math> is a multiple of 3 or not, yields a sequence with

:<math> \delta(N) \leq \log_9(N)+1 </math>

for sufficiently large <math> N </math>. Currently, this is the record-holder for slow growing discrepancy.

The Thue-Morse sequence has discrepancy <math> \delta(N) \gg N^{\log_4(3)} </math>. (See the discussion following [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4775 this comment] and the next one for an easy proof of <math>\gg N^{1/2}</math>, and the paper of Newman for the discrepancy along the 3-HAP.)

A random sequence (each <math> x_i </math> chosen independently) has discrepancy at least (asymptotically) <math> \sqrt{2N \log\log N} </math> by the law of the iterated logarithm.

Determining if the discrepancy of [http://en.wikipedia.org/wiki/Liouville_function Liouville's lambda function] is <math> O(n^{1/2+\epsilon})</math> is equivalent to solving the Riemann hypothesis. However, this growth cannot be bounded above by <math>n^{1/2-\epsilon}</math> for any positive ε.

==Interesting subquestions==

Given the length of time that the Erdős discrepancy problem has been open, the chances that it will be solved by Polymath5 are necessarily small. However, there are a number of interesting questions that we do not know the answers to, several of which have arisen naturally from the experimental evidence. Good answers to some of these would certainly constitute publishable results. Here is a partial list -- further additions are very welcome. (When the more theoretical part of the project starts, this section will probably grow substantially and become a separate page.)

*Is there an infinite sequence of discrepancy 2? (Given how long a finite sequence can be, it seems unlikely that we could answer this question just by a clever search of all possibilities on a computer.)

*The long sequences of low discrepancy discovered by computer all have some kind of approximate weak multiplicativity. If we take a hypothetical counterexample <math>(x_n)</math> (which we could even assume has discrepancy 2), can we prove that it, or some other counterexample derived from it by passing to HAP-subsequences and taking pointwise limits, has some kind of interesting multiplicative structure?

*Closely related to the previous question. If <math>(x_n)</math> is a hypothetical counterexample, must it satisfy a compactness property of the following kind: for every positive c there exists M such that if you take any M HAP-subsequences, then there must be two of them, <math>(y_n)</math> and <math>(z_n)</math>, such that <math>\lim\inf N^{-1}\sum_{n=1}^Ny_nz_n\geq 1-c</math>?

*An even weaker question. If <math>(x_n)</math> is a hypothetical counterexample, must it have two HAP-subsequences <math>(y_n)</math> and <math>(z_n)</math> such that the lim inf of <math>N^{-1}\sum_{n=1}^N y_nz_n</math> is greater than 0?

*A similar question, perhaps equivalent to the previous one (this should be fairly easy to check). Given a sequence <math>(x_n)</math> define f(N) to be the average of <math>x_ax_bx_cx_d</math> over all quadruples (a,b,c,d) such that ab=cd and <math>a,b,c,d\leq N</math>. If <math>(x_n)</math> is a counterexample, does that imply that <math>\lim\inf f(N)</math> is greater than 0? [http://thomas1111.wordpress.com/2010/01/15/average-over-quadruples-of-the-first-1124-sequence/ See here for a computation of this average for the first 1124 sequence].

*Is there any completely multiplicative counterexample? (This may turn out to be a very hard question. If so, then answering the previous questions could turn out to be the best we can hope for without making a substantial breakthrough in analytic number theory.)

*Does there exist a <math>\pm 1</math> sequence such that the corresponding discrepancy function grows at a rate that is slower than <math>c\log n</math> for any positive c?

*Does the number of sequences of length n and discrepancy at most C grow exponentially with n or more slowly than exponentially? (Obviously if the conjecture is true then it must be zero for large enough n, but the hope is that this question is a more realistic initial target.)

==General proof strategies==

This section contains links to other pages in which potential approaches to solving the problem are described.

[[First obtain multiplicative structure and then obtain a contradiction]]

[[Find a different parameter, show that it tends to infinity, and show that that implies that the discrepancy tends to infinity]]

[[Prove the result for shifted HAPs instead of HAPs]]

==Annotated Bibliography==

*Blog Discussion on Gowers's Weblog
**[http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/ Zeroth Post] (Dec 17 - Jan 6).
**[http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/ First Post] (Jan 6 - Jan 12).
**[http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/ Second Post] (Jan 9 - Jan 11).
**[http://gowers.wordpress.com/2010/01/11/the-erds-discrepancy-problem-iii/ Third Post] (Jan 11 - Jan 14).
**[http://gowers.wordpress.com/2010/01/14/the-erds-discrepancy-problem-iv/ Fourth Post] (Jan 14 - 16).
**[http://gowers.wordpress.com/2010/01/16/the-erds-discrepancy-problem-v/ Fifth Post] (Jan 16-19).
**[http://gowers.wordpress.com/2010/01/19/edp1-the-official-start-of-polymath5/ First Theoretical Post] (Jan 19-21)
**[http://gowers.wordpress.com/2010/01/21/edp2-a-few-lessons-from-edp1/ Second Theoretical Post] (Jan 21-26)
**[http://gowers.wordpress.com/2010/01/26/edp3-a-very-brief-report-on-where-we-are/ Third Theoretical Post] (Jan 26 -?)
**[http://gowers.wordpress.com/2010/01/30/edp4-focusing-on-multiplicative-functions/ Fourth Theoretical Post] (Jan 30- Feb 2)
**[http://gowers.wordpress.com/2010/02/02/edp5-another-very-brief-summary/ Fifth Theoretical Post] (Feb 2 - ?)

*Mathias, A. R. D. [http://www.dpmms.cam.ac.uk/~ardm/erdoschu.pdf On a conjecture of Erdős and Čudakov]. Combinatorics, geometry and probability (Cambridge, 1993).

This one page paper establishes that the maximal length of sequence for the case where C=1 is 11, and is the starting point for our experimental studies.

*Tchudakoff, N. G. Theory of the characters of number semigroups. J. Indian Math. Soc. (N.S.) 20 (1956), 11--15. MR0083515 (18,719e)

The Mathias paper states that one of Erdős's problem lists states that this paper "studies related questions". The Math Review for this paper states that it summarizes results from seven other papers that are in Russian. The Math Review leaves the impression that the topic concerns characters of modulus 1.

*Borwein, Peter, and Choi, Stephen. [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P204.pdf A variant of Liouville's lambda function: some surprizing formulae].

Explicit formulas for the discrepancy of some completely multiplicative functions whose discrepancy is logarithmic. A typical example is: Let <math>\lambda_3(n) = (-1)^{\omega_3(n)}</math> where <math>\omega_3(n)</math> is the number of distinct prime factors congruent to <math>-1 \bmod 3</math> in <math>n</math> (with multiple factors counted multiply). Then the discrepancy of the <math>\lambda_3</math> sequence up to <math>N</math> is exactly the number of 1's in the base three expansion of <math>N</math>. (This turns out not to be so surprising after all, since <math>\lambda_3(n)</math> is precisely the same as the ternary function defined in the second item of the Simple Observations section.)

*Borwein, Peter, Choi, Stephen and Coons, Michael. [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P225.pdf Completely multiplicative functions taking values in {-1,1}]. The published version of the above paper, which explains the results and proofs more clearly than the preprint, and is more explicit about the relationship with Erdős's question.

*Granville and Soundararajan. [http://arxiv.org/abs/math/0702389 Multiplicative functions in arithmetic progressions]

In this [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4729 blog post] Terry Tao wrote: "Granville and Soundararajan have made some study of the discrepancy of bounded multiplicative functions. The situation is remarkably delicate and number-theoretical (and is closely tied with the Granville-Soundararajan theory of pretentious characters)."

*Wirsing, E. Das asymptotische Verhalten von Summen über multiplikative Funktionen. II. (German) [The asymptotic behavior of sums of multiplicative functions. II.] Acta Math. Acad. Sci. Hungar. 18 1967 411--467. MR0223318 (36 #6366) [http://michaelnielsen.org/polymath1/index.php?title=Wirsing_translation (partial) English translation]

*Newman, D. J. [http://www.jstor.org/stable/2036455 On the number of binary digits in a multiple of three]. Proc Amer Math Soc. 21(1969): 719--721.
Proves that the Thue-Morse sequence has discrepancy <math> \gg N^{\log_4(3)}</math>

*Halász, G. [http://gowers.wordpress.com/2010/02/02/edp5-another-very-brief-summary/#comment-5806 On random multiplicative functions]. Hubert Delange colloquium (Orsay, 1982), 74–96, Publ. Math. Orsay, 83-4, Univ. Paris XI, Orsay, 1983.

The discrepancy of a random multiplicative function is close to $\sqrt{N}$.

=== Problem lists on which this problem appears ===

*Erdős, P. and Graham, R. [http://www.math.ucsd.edu/~fan/ron/papers/79_09_combinatorial_number_theory.pdf Old and New Problems and Results in Combinatorial Number Theory: van der Waerden's Theorem and Related Topics], L'Enseignement Math. 25 (1979), 325-344.

Our problem is mentioned at the bottom of page 331, where they indicate knowledge of a coloring with logarithmic discrepancy. On pages 330-1, they review work on the non-homogeneous problem.

*As [http://www.math.niu.edu/~rusin/known-math/93_back/prizes.erd this web page] reveals, the Erdős discrepancy problem was a $500-dollar problem of Erdős, so it is clear that he regarded it as pretty hard.

*P. Erdős. [http://www.renyi.hu/~p_erdos/1957-13.pdf Some unsolved problems], Michigan Math. J. 4 (1957), 291--300 MR20 #5157; Zentralblatt 81,1.

Problem 9 of this paper is ours. Erdős dates the problem to around 1932, and notes what we know about Liouville's function (lower and upper bound).

*Finch, S. [http://algo.inria.fr/csolve/ec.pdf Two-colorings of positive integers]. Dated May 27, 2008.

Summarizes knowledge of discrepancy of two colorings when the discrepancy is restricted to homogeneous progressions, nonhomogeneous progressions, and homogeneous quasi-progressions. Contains bibliography with 17 entries, including most of those above.

=== non-homogeneous AP, quasi-AP, and other related discrepancy papers ===

*Hochberg, Robert. [http://www.springerlink.com/content/q02424284373qw45/ Large Discrepancy In Homogenous Quasi-Arithmetic Progressions]. Combinatorica, Volume 26 (2006), Number 1.

Roth's method for the <math> n^{1/4}</math> lower bound on nonhomogeneous AP discrepancy is adapted to give a lower bound for homogeneous quasi-AP discrepancy. This result is weaker than the Vijay result, but uses a different method.

*Vijay, Sujith. [http://www.combinatorics.org/Volume_15/PDF/v15i1r104.pdf On the discrepancy of quasi-progressions]. Electronic Journal of Combinatorics, R104 of Volume 15(1), 2008.

A quasi-progression is a sequence of the form <math>\lfloor k \alpha \rfloor </math>, with <math>1\leq k \leq K</math> for some K. The main theorem of interest is: If the integers from 0 to n are 2-coloured, there exists a quasi-progression that has discrepancy at least <math> (1/50)n^{1/6}</math>.

* Doerr, Benjamin; Srivastav, Anand; Wehr, Petra. [http://www.combinatorics.org/Volume_11/Abstracts/v11i1r5.html Discrepancy of Cartesian products of arithmetic progressions]. Electron. J. Combin. 11 (2004), no. 1, Research Paper 5, 16 pp. (electronic).

Abstract: We determine the combinatorial discrepancy of the hypergraph H of cartesian
products of d arithmetic progressions in the <math> [N]^d</math> –lattice.
The study of such higher dimensional arithmetic progressions is motivated by a
multi-dimensional version of van derWaerden’s theorem, namely the Gallai-theorem
(1933). We solve the discrepancy problem for d–dimensional arithmetic progressions
by proving <math>disc(H) = \Theta�(N^{d/4})</math> for every fixed positive integer d. This extends the famous
lower bound of <math> \Omega(N^{1/4})</math> of Roth (1964) and the matching upper bound <math> O(N^{1/4})</math>
of Matouˇsek and Spencer (1996) from d = 1 to arbitrary, fixed d. To establish
the lower bound we use harmonic analysis on locally compact abelian groups. For
the upper bound a product coloring arising from the theorem of Matouˇsek and
Spencer is sufficient. We also regard some special cases, e.g., symmetric arithmetic
progressions and infinite arithmetic progressions.

*Cilleruelo, Javier; Hebbinghaus, Nils [http://www.ams.org/mathscinet-getitem?mr=2547932 Discrepancy in generalized arithmetic progressions]. European J. Combin. 30 (2009), no. 7, 1607--1611. MR 2547932

*Valkó, Benedek. [Discrepancy of arithmetic progressions in higher dimensions]. (English summary)
J. Number Theory 92 (2002), no. 1, 117--130. MR1880588 (2003b:11071)

Abstract:
K. F. Roth (1964, Acta. Arith.9, 257–260) proved that the discrepancy of arithmetic progressions contained in [N] is at least <math> cN^{1/4} </math>, and later it was proved that this result is sharp. We consider the d-dimensional version of this problem. We give a lower estimate for the discrepancy of arithmetic progressions on <math> [N]^d </math> and prove that this result is nearly sharp. We use our results to give an upper estimate for the discrepancy of lines on an N×N lattice, and we also give an estimate for the discrepancy of a related random hypergraph.

*Roth, K. F., Remark concerning integer sequences. Acta Arith. 9 1964 257--260. MR0168545 (29 #5806)

Shows that the discrepancy on APs is at least <math>cN^{1/4}</math>.

Multiplicative sequences

2010-02-03T21:48:39Z

Thomas: improved wording

Computer proof that completely multiplicative sequences have discrepancy greater than 2

2010-02-03T16:59:20Z

Thomas: fixed HTML tags

The following [[multiplicative.c|C code]] implements [[algorithm for finding multiplicative sequences with bounded discrepancy|this algorithm]] for making deductions about multiplicative functions of discrepancy 2. At any given time, the "knowledge" that this algorithm has is given by:

* a reduced expression for f(n) for each n, of the form f(n) = f(m) for some |m| \leq n (where we adopt the convention that f is odd, i.e. f(-m) = -f(m)). Thus for instance if we know that f(2)=-1, then we can reduce f(12)=f(-3). This is recorded by an array vals, thus for instance vals[12]=-3 would be recording the fact that f(12)=f(-3).

* Upper and lower bounds for each of the partial sums f[1,n] := f(1)+...+f(n).

The "easy" deductions that the algorithm makes automatically are:

* consequences of multiplicativity (e.g. f(n^2)=1);
* Adjusting upper and lower bounds for f[1,n] based on knowledge of f[1,n-1] and f(n), or of f[1,n+1] and f(n+1);
* Deducing the value of f(n) given enough information about f[1,n] and f[1,n-1]

The format of the program is

: [program name] <math>n_1</math> <math>n_2</math> ...

which makes the assumptions <math>f(n_1)=f(n_2)=\ldots=1</math> and sees what one can deduce from there. For instance, "[program name] 2 -3" would assume that f(2)=1 and f(3)=-1.

Currently the computer only looks at f(n) for n < SIZE, where SIZE by default is 400. But one could certainly enlarge SIZE which in principle would make the computer "smarter".

Below are some examples of how this code works.

== With no assumptions ==

[program name]

Deducing...
Deducing...

| 0 | 1 2 3 1 5 6 7 2 1 [0-9]
10 11 3 13 14 15 1 17 2 19 [10-19]
5 21 22 23 6 1 26 3 7 29 [20-29]
30 31 2 33 34 35 1 37 38 39 [30-39]
10 41 42 43 11 5 46 47 3 1 [40-49]
2 51 13 53 6 55 14 57 58 59 [50-59]
15 61 62 7 1 65 66 67 17 69 [60-69]
70 71 2 73 74 3 19 77 78 79 [70-79]
5 1 82 83 21 85 86 87 22 89 [80-89]
10 91 23 93 94 95 6 97 2 11 [90-99]
1 101 102 103 26 105 106 107 3 109 [100-109]
110 111 7 113 114 115 29 13 118 119 [110-119]
30 1 122 123 31 5 14 127 2 129 [120-129]
130 131 33 133 134 15 34 137 138 139 [130-139]
35 141 142 143 1 145 146 3 37 149 [140-149]
6 151 38 17 154 155 39 157 158 159 [150-159]
10 161 2 163 41 165 166 167 42 1 [160-169]
170 19 43 173 174 7 11 177 178 179 [170-179]
5 181 182 183 46 185 186 187 47 21 [180-189]
190 191 3 193 194 195 1 197 22 199 [190-199]

== Assuming f(2)=+1 ==

[program name] 2

Setting f(2)=1...
Deducing...
Setting f(3)=-1...
Deducing...
Setting f(5)=-1...
Deducing...
Setting f(7)=-1...
Deducing...
Deducing...

| 0 | 1 1 ^ -1 1 ^ -1 -1 | -1 1 | 1 [0-9]
-1 | 11 -1 13 -1 1 1 17 1 19 [10-19]
-1 1 11 23 -1 1 13 -1 -1 29 [20-29]
1 31 1 -11 17 1 1 37 19 -13 [30-39]
-1 41 1 43 11 -1 23 47 -1 1 [40-49]
1 -17 13 53 -1 -11 -1 -19 29 59 [50-59]
1 61 31 -1 1 -13 -11 67 17 -23 [60-69]
1 71 1 73 37 -1 19 -11 -13 79 [70-79]
-1 1 41 83 1 -17 43 -29 11 89 [80-89]
-1 -13 23 -31 47 -19 -1 97 1 11 [90-99]
1 101 -17 103 13 -1 53 107 -1 109 [100-109]
-11 -37 -1 113 -19 -23 29 13 59 -17 [110-119]
1 | 1 61 -41 31 -1 -1 127 1 -43 [120-129]
-13 131 -11 -19 67 1 17 137 -23 139 [130-139]
1 -47 71 143 1 -29 73 -1 37 149 [140-149]
-1 151 19 17 -11 -31 -13 157 79 -53 [150-159]
-1 -23 1 163 41 11 83 167 1 | 1 [160-169]
-17 19 43 173 -29 -1 11 -59 89 179 [170-179]
-1 181 -13 -61 23 -37 -31 187 47 1 [180-189]
-19 191 -1 193 97 13 1 197 11 199 [190-199]

Note that f[15,19] = 3 + f(17)+f(19) and f[168,171]=2-f(17)+f(19), giving a nontrivial deduction that f(19)=-1. We insert this information:

[program name] 2 -19

Setting f(2)=1...
Setting f(19)=-1...
Deducing...
Setting f(3)=-1...
Deducing...
Setting f(5)=-1...
Deducing...
Setting f(7)=-1...
Deducing...
Deducing...

| 0 | 1 1 ^ -1 1 ^ -1 -1 | -1 1 | 1 [0-9]
-1 | 11 -1 13 -1 1 1 17 1 -1 [10-19]
-1 1 11 23 -1 1 13 -1 -1 29 [20-29]
1 31 1 -11 17 1 1 37 -1 -13 [30-39]
-1 41 1 43 11 -1 23 47 -1 1 [40-49]
1 -17 13 53 -1 -11 -1 1 29 59 [50-59]
1 61 31 -1 1 -13 -11 67 17 -23 [60-69]
1 71 1 73 37 -1 -1 -11 -13 79 [70-79]
-1 1 41 83 1 -17 43 -29 11 89 [80-89]
-1 -13 23 -31 47 1 -1 97 1 11 [90-99]
1 101 -17 103 13 -1 53 107 -1 109 [100-109]
-11 -37 -1 113 1 -23 29 13 59 -17 [110-119]
1 | 1 61 -41 31 -1 -1 127 1 -43 [120-129]
-13 131 -11 1 67 1 17 137 -23 139 [130-139]
1 -47 71 143 1 -29 73 -1 37 149 [140-149]
-1 151 -1 17 -11 -31 -13 157 79 -53 [150-159]
-1 -23 1 163 41 11 83 167 1 | 1 [160-169]
-17 -1 43 173 -29 -1 11 -59 89 179 [170-179]
-1 181 -13 -61 23 -37 -31 187 47 1 [180-189]
1 191 -1 193 97 13 1 197 11 199 [190-199]

== Assuming f(2)=f(37)=+1 ==

[program name] 2 -19 37

Setting f(2)=1...
Setting f(19)=-1...
Setting f(37)=1...
Deducing...
Setting f(3)=-1...
Deducing...
Setting f(5)=-1...
Deducing...
Setting f(7)=-1...
Deducing...
Setting f(17)=-1...
Deducing...
Setting f(11)=1...
Setting f(59)=-1...
Setting f(151)=1...
Deducing...
Setting f(31)=-1...
Setting f(29)=1...
Setting f(101)=-1...
Setting f(109)=1...
Setting f(113)=1...
Deducing...
Setting f(61)=1...
Setting f(103)=-1...
Setting f(13)=-1...
Setting f(127)=1...
Deducing...
Setting f(23)=1...
Setting f(53)=1...
Setting f(41)=1...
Deducing...
Setting f(43)=-1...
Setting f(47)=-1...
Setting f(67)=1...
Setting f(83)=-1...
Setting f(89)=-1...
Setting f(107)=1...
Setting f(79)=1...
Setting f(163)=-1...
Deducing...
Contradiction! -1 >= f[1,47] >= 1

| 0 | 1 1 ^ -1 1 ^ -1 -1 | -1 1 | 1 [0-9]
-1 | 1 -1 | -1 -1 v 1 1 | -1 1 | -1 [10-19]
-1 v 1 1 | 1 -1 | 1 -1 | -1 -1 v 1 [20-29]
1 | -1 1 | -1 -1 v 1 1 | 1 -1 | 1 [30-39]
-1 | 1 1 ^ -1 1 ^ -1 1 ^ -1 -1 v 1 [40-49]
1 | 1 -1 | 1 -1 | -1 -1 v 1 1 | -1 [50-59]
1 | 1 -1 | -1 1 | 1 -1 | 1 -1 | -1 [60-69]
1 | 71 1 73 1 ^ -1 -1 | -1 1 | 1 [70-79]
-1 1 1 -1 1 | 1 -1 -1 1 -1 [80-89]
-1 v 1 1 | 1 -1 1 -1 97 1 | 1 [90-99]
1 ^ -1 1 ^ -1 -1 | -1 1 | 1 -1 | 1 [100-109]
-1 | -1 -1 v 1 1 | -1 1 | -1 -1 v 1 [110-119]
1 | 1 1 ^ -1 -1 | -1 -1 v 1 1 | 1 [120-129]
1 131 -1 1 1 1 -1 137 -1 139 [130-139]
1 1 71 -1 1 -1 73 -1 1 149 [140-149]
-1 | 1 -1 | -1 -1 v 1 1 | 157 1 ^ -1 [150-159]
-1 | -1 1 | -1 1 | 1 -1 167 1 | 1 [160-169]
1 ^ -1 -1 173 -1 | -1 1 | 1 -1 179 [170-179]
-1 181 1 -1 1 -1 1 -1 -1 1 [180-189]
1 191 -1 193 97 -1 1 197 1 199 [190-199]

== Assuming f(2)=+1; f(37)=-1 ==

[program name] 2 -19 37

Setting f(2)=1...
Setting f(19)=-1...
Setting f(37)=-1...
Deducing...
Setting f(3)=-1...
Deducing...
Setting f(5)=-1...
Setting f(77)=1...
Deducing...
Setting f(7)=-1...
f(11)=f(77)/f(11)=-1... Setting f(11)=-1...
Deducing...
Setting f(13)=1...
Setting f(31)=-1...
Setting f(17)=-1...
Setting f(67)=-1...
Deducing...
Setting f(23)=1...
Setting f(41)=1...
Setting f(53)=-1...
Contradiction! f(1) = 1 and f(1) = -1 (from f(69)=1)

| 0 | 1 1 ^ -1 1 ^ -1 -1 | -1 1 | 1 [0-9]
-1 | -1 -1 v 1 -1 v 1 1 | -1 1 | -1 [10-19]
-1 v 1 -1 v 1 -1 1 1 -1 -1 29 [20-29]
1 | -1 1 | 1 -1 | 1 1 ^ -1 -1 | -1 [30-39]
-1 v 1 1 43 -1 | -1 1 47 -1 v 1 [40-49]
1 | 1 1 ^ -1 -1 1 -1 1 29 59 [50-59]
1 61 -1 | -1 1 | -1 1 | -1 -1 v -1 [60-69]
1 71 1 73 -1 | -1 -1 v 1 -1 v 79 [70-79]
-1 1 1 83 1 | 1 43 -29 -1 89 [80-89]
-1 | -1 1 1 47 1 -1 97 1 -1 [90-99]
1 101 1 103 1 -1 -1 107 -1 109 [100-109]
1 | 1 -1 | 113 1 -1 29 1 59 v 1 [110-119]
1 | 1 61 -1 -1 | -1 -1 v 127 1 -43 [120-129]
-1 131 1 | 1 -1 | 1 -1 | 137 -1 139 [130-139]
1 -47 71 -1 1 -29 73 -1 -1 149 [140-149]
-1 151 -1 | -1 1 | 1 -1 | 157 79 1 [150-159]
-1 -1 1 163 1 -1 83 167 1 | 1 [160-169]
1 ^ -1 43 173 -29 -1 -1 -59 89 179 [170-179]
-1 181 -1 -61 1 v 1 1 | 1 47 1 [180-189]
1 191 -1 193 97 1 1 197 -1 199 [190-199]

== Using f(2)=-1 ==

[program name] -2

Setting f(2)=-1...
Deducing...
Deducing...

| 0 | 1 -1 | 3 1 5 -3 7 -1 1 [0-9]
-5 11 3 13 -7 15 1 17 -1 19 [10-19]
5 21 -11 23 -3 1 -13 3 7 29 [20-29]
-15 31 -1 33 -17 35 1 37 -19 39 [30-39]
-5 41 -21 43 11 5 -23 47 3 1 [40-49]
-1 51 13 53 -3 55 -7 57 -29 59 [50-59]
15 61 -31 7 1 65 -33 67 17 69 [60-69]
-35 71 -1 73 -37 3 19 77 -39 79 [70-79]
5 1 -41 83 21 85 -43 87 -11 89 [80-89]
-5 91 23 93 -47 95 -3 97 -1 11 [90-99]
1 101 -51 103 -13 105 -53 107 3 109 [100-109]
-55 111 7 113 -57 115 29 13 -59 119 [110-119]
-15 1 -61 123 31 5 -7 127 -1 129 [120-129]
-65 131 33 133 -67 15 -17 137 -69 139 [130-139]
35 141 -71 143 1 145 -73 3 37 149 [140-149]
-3 151 -19 17 -77 155 39 157 -79 159 [150-159]
-5 161 -1 163 41 165 -83 167 -21 1 [160-169]
-85 19 43 173 -87 7 11 177 -89 179 [170-179]
5 181 -91 183 -23 185 -93 187 47 21 [180-189]
-95 191 3 193 -97 195 1 197 -11 199 [190-199]

== Assuming f(5)=+1 ==

[program name] -2 5

Setting f(2)=-1...
Setting f(5)=1...
Deducing...
Setting f(3)=-1...
Deducing...
Setting f(7)=-1...
Setting f(241)=1...
Deducing...
Deducing...

| 0 | 1 -1 | -1 1 | 1 1 ^ -1 -1 | 1 [0-9]
-1 | 11 -1 13 1 -1 1 17 -1 19 [10-19]
1 | 1 -11 23 1 | 1 -13 -1 -1 29 [20-29]
1 31 -1 -11 -17 -1 1 37 -19 -13 [30-39]
-1 41 -1 43 11 1 -23 47 -1 1 [40-49]
-1 -17 13 53 1 11 1 -19 -29 59 [50-59]
-1 61 -31 -1 1 13 11 67 17 -23 [60-69]
1 71 -1 73 -37 -1 19 -11 13 79 [70-79]
1 | 1 -41 83 1 17 -43 -29 -11 89 [80-89]
-1 -13 23 -31 -47 19 1 97 -1 11 [90-99]
1 101 17 103 -13 1 -53 107 -1 109 [100-109]
-11 -37 -1 113 19 23 29 13 -59 -17 [110-119]
1 | 1 -61 -41 31 1 1 127 -1 -43 [120-129]
-13 131 -11 -19 -67 -1 -17 137 23 139 [130-139]
-1 -47 -71 143 1 29 -73 -1 37 149 [140-149]
1 151 -19 17 11 31 -13 157 -79 -53 [150-159]
-1 -23 -1 163 41 -11 -83 167 -1 1 [160-169]
-17 19 43 173 29 -1 11 -59 -89 179 [170-179]
1 181 13 -61 -23 37 31 187 47 1 [180-189]
-19 191 -1 193 -97 -13 1 197 -11 199 [190-199]
-1 -67 -101 -29 -17 41 -103 23 13 209 [200-209]
-1 211 53 -71 -107 43 1 -31 -109 -73 [210-219]
11 221 37 223 1 | 1 -113 227 -19 229 [220-229]
-23 11 -29 233 -13 47 59 -79 17 239 [230-239]
-1 | 1 -1 | -1 61 1 41 247 -31 -83 [240-249]
-1 251 -1 253 -127 -17 1 257 43 -37 [250-259]
13 29 -131 263 11 53 19 -89 67 269 [260-269]
1 271 17 13 -137 11 -23 277 -139 31 [270-279]
1 281 47 283 71 -19 -143 -41 -1 1 [280-289]
-29 -97 73 293 1 59 -37 -11 -149 299 [290-299]
-1 -43 -151 -101 19 61 -17 307 -11 -103 [300-309]
-31 311 13 313 -157 -1 79 317 53 319 [310-319]
1 -107 23 323 1 13 -163 -109 -41 -47 [320-329]
11 331 83 37 -167 67 1 337 -1 -113 [330-339]
17 341 -19 -1 -43 -23 -173 347 -29 349 [340-349]
1 -13 -11 353 59 71 89 17 -179 359 [350-359]
-1 1 -181 -1 -13 73 61 367 23 41 [360-369]
-37 -53 -31 373 -187 -1 -47 377 -1 379 [370-379]
19 -127 -191 383 1 -11 -193 43 97 389 [380-389]
13 391 -1 -131 -197 79 11 397 -199 19 [390-399]

== Assuming f(5)=f(11)= 1 ==

[program name] -2 5 11

Setting f(2)=-1...
Setting f(5)=1...
Setting f(11)=1...
Deducing...
Setting f(3)=-1...
Deducing...
Setting f(7)=-1...
Setting f(241)=1...
Deducing...
Setting f(23)=-1...
Setting f(17)=-1...
Setting f(19)=1...
Deducing...
Setting f(13)=-1...
Setting f(47)=-1...
Setting f(59)=1...
Setting f(67)=-1...
Setting f(239)=1...
Setting f(251)=1...
Setting f(31)=1...
Deducing...
Setting f(29)=-1...
Setting f(53)=-1...
Contradiction! f(1) = 1 and f(1) = -1 (from f(58)=-1)

| 0 | 1 -1 | -1 1 | 1 1 ^ -1 -1 | 1 [0-9]
-1 | 1 -1 | -1 1 | -1 1 | -1 -1 v 1 [10-19]
1 | 1 -1 | -1 1 | 1 1 ^ -1 -1 | -1 [20-29]
1 | 1 -1 | -1 1 | -1 1 | 37 -1 1 [30-39]
-1 41 -1 43 1 | 1 1 ^ -1 -1 | 1 [40-49]
-1 | 1 -1 | -1 1 | 1 1 ^ -1 1 1 [50-59]
-1 61 -1 | -1 1 | -1 1 | -1 -1 v 1 [60-69]
1 | 71 -1 73 -37 -1 1 -1 -1 79 [70-79]
1 | 1 -41 83 1 -1 -43 1 -1 89 [80-89]
-1 | 1 -1 | -1 1 | 1 1 ^ 97 -1 1 [90-99]
1 101 -1 103 1 | 1 1 107 -1 109 [100-109]
-1 -37 -1 113 1 -1 -1 | -1 -1 v 1 [110-119]
1 | 1 -61 -41 1 | 1 1 ^ 127 -1 -43 [120-129]
1 131 -1 | -1 1 | -1 1 | 137 -1 139 [130-139]
-1 1 -71 -1 1 -1 -73 -1 37 149 [140-149]
1 151 -1 | -1 1 | 1 1 ^ 157 -79 1 [150-159]
-1 1 -1 163 41 -1 -83 167 -1 v 1 [160-169]
1 | 1 43 173 -1 -1 1 -1 -89 179 [170-179]
1 181 -1 -61 1 37 1 -1 -1 1 [180-189]
-1 191 -1 193 -97 1 1 197 -1 199 [190-199]
-1 1 -101 1 1 41 -103 -1 -1 1 [200-209]
-1 211 -1 -71 -107 43 1 -1 -109 -73 [210-219]
1 | 1 37 223 1 | 1 -113 227 -1 229 [220-229]
1 | 1 1 233 1 -1 1 -79 -1 | 1 [230-239]
-1 | 1 -1 | -1 61 1 41 -1 -1 -83 [240-249]
-1 | 1 -1 | -1 -127 1 1 257 43 -37 [250-259]
-1 -1 -131 263 1 -1 1 -89 -1 269 [260-269]
1 271 -1 | -1 -137 1 1 277 -139 1 [270-279]
1 281 -1 283 71 -1 1 -41 -1 1 [280-289]
1 -97 73 293 1 | 1 -37 -1 -149 1 [290-299]
-1 -43 -151 -101 1 61 1 307 -1 -103 [300-309]
-1 311 -1 313 -157 -1 79 317 -1 -1 [310-319]
1 -107 -1 | -1 1 | -1 -163 -109 -41 1 [320-329]
1 331 83 37 -167 -1 1 337 -1 -113 [330-339]
-1 | 1 -1 | -1 -43 1 -173 347 1 349 [340-349]
1 | 1 -1 | 353 1 71 89 -1 -179 359 [350-359]
-1 1 -181 -1 1 73 61 367 -1 41 [360-369]
-37 1 -1 373 1 -1 1 1 -1 379 [370-379]
1 -127 -191 383 1 -1 -193 43 97 389 [380-389]
-1 1 -1 -131 -197 79 1 397 -199 1 [390-399]

== Back to assuming f(5) = 1 ==

From the previous section we can now assume f(11)=-1.

[program name] -2 5 -11

Setting f(2)=-1...
Setting f(5)=1...
Setting f(11)=-1...
Deducing...
Setting f(3)=-1...
Deducing...
Setting f(7)=-1...
Setting f(13)=1...
Setting f(23)=-1...
Setting f(241)=1...
Deducing...
Contradiction! -1 >= f[1,23] >= 1

| 0 | 1 -1 | -1 1 | 1 1 ^ -1 -1 | 1 [0-9]
-1 | -1 -1 v 1 1 | -1 1 | 17 -1 19 [10-19]
1 1 1 ^ -1 1 | 1 -1 -1 -1 29 [20-29]
1 31 -1 1 -17 -1 1 37 -19 -1 [30-39]
-1 41 -1 43 -1 1 1 47 -1 1 [40-49]
-1 -17 1 53 1 -1 1 -19 -29 59 [50-59]
-1 61 -31 -1 1 1 -1 67 17 1 [60-69]
1 71 -1 73 -37 -1 19 1 1 79 [70-79]
1 | 1 -41 83 1 17 -43 -29 1 89 [80-89]
-1 -1 -1 -31 -47 19 1 97 -1 | -1 [90-99]
1 | 101 17 103 -1 1 -53 107 -1 109 [100-109]
1 -37 -1 113 19 -1 29 1 -59 -17 [110-119]
1 | 1 -61 -41 31 1 1 127 -1 -43 [120-129]
-1 131 1 -19 -67 -1 -17 137 -1 139 [130-139]
-1 -47 -71 -1 1 29 -73 -1 37 149 [140-149]
1 151 -19 17 -1 31 -1 157 -79 -53 [150-159]
-1 1 -1 163 41 1 -83 167 -1 1 [160-169]
-17 19 43 173 29 -1 -1 -59 -89 179 [170-179]
1 181 1 -61 1 37 31 -17 47 1 [180-189]
-19 191 -1 193 -97 -1 1 197 1 199 [190-199]
-1 -67 -101 -29 -17 41 -103 -1 1 -19 [200-209]
-1 211 53 -71 -107 43 1 -31 -109 -73 [210-219]
-1 17 37 223 1 1 -113 227 -19 229 [220-229]
1 -1 -29 233 -1 47 59 -79 17 239 [230-239]
-1 1 -1 | -1 61 1 41 19 -31 -83 [240-249]
-1 251 -1 1 -127 -17 1 257 43 -37 [250-259]
1 29 -131 263 -1 53 19 -89 67 269 [260-269]
1 271 17 1 -137 -1 1 277 -139 31 [270-279]
1 281 47 283 71 -19 1 -41 -1 1 [280-289]
-29 -97 73 293 1 59 -37 1 -149 -1 [290-299]
-1 -43 -151 -101 19 61 -17 307 1 -103 [300-309]
-31 311 1 313 -157 -1 79 317 53 -29 [310-319]
1 -107 -1 323 1 1 -163 -109 -41 -47 [320-329]
-1 331 83 37 -167 67 1 337 -1 -113 [330-339]
17 -31 -19 -1 -43 1 -173 347 -29 349 [340-349]
1 -1 1 353 59 71 89 17 -179 359 [350-359]
-1 1 -181 -1 -1 73 61 367 -1 41 [360-369]
-37 -53 -31 373 17 -1 -47 29 -1 379 [370-379]
19 -127 -191 383 1 1 -193 43 97 389 [380-389]
1 -17 -1 -131 -197 79 -1 397 -199 19 [390-399]

== No assumptions again ==

[program name] -2 -5

Setting f(2)=-1...
Setting f(5)=-1...
Deducing...
Deducing...

| 0 | 1 -1 | 3 1 -1 -3 7 -1 1 [0-9]
1 11 3 13 -7 -3 1 17 -1 19 [10-19]
-1 21 -11 23 -3 1 -13 3 7 29 [20-29]
3 31 -1 33 -17 -7 1 37 -19 39 [30-39]
1 41 -21 43 11 -1 -23 47 3 1 [40-49]
-1 51 13 53 -3 -11 -7 57 -29 59 [50-59]
-3 61 -31 7 1 -13 -33 67 17 69 [60-69]
7 71 -1 73 -37 3 19 77 -39 79 [70-79]
-1 1 -41 83 21 -17 -43 87 -11 89 [80-89]
1 91 23 93 -47 -19 -3 97 -1 11 [90-99]
1 101 -51 103 -13 -21 -53 107 3 109 [100-109]
11 111 7 113 -57 -23 29 13 -59 119 [110-119]
3 1 -61 123 31 -1 -7 127 -1 129 [120-129]
13 131 33 133 -67 -3 -17 137 -69 139 [130-139]
-7 141 -71 143 1 -29 -73 3 37 149 [140-149]
-3 151 -19 17 -77 -31 39 157 -79 159 [150-159]
1 161 -1 163 41 -33 -83 167 -21 1 [160-169]
17 19 43 173 -87 7 11 177 -89 179 [170-179]
-1 181 -91 183 -23 -37 -93 187 47 21 [180-189]
19 191 3 193 -97 -39 1 197 -11 199 [190-199]
-1 201 -101 203 51 -41 -103 23 13 209 [200-209]
21 211 53 213 -107 -43 -3 217 -109 219 [210-219]
-11 221 -111 223 -7 1 -113 227 57 229 [220-229]
23 231 -29 233 -13 -47 59 237 -119 239 [230-239]
-3 241 -1 3 61 -1 -123 247 -31 249 [240-249]
1 251 7 253 -127 -51 1 257 -129 259 [250-259]
-13 29 -131 263 -33 -53 -133 267 67 269 [260-269]
3 271 17 273 -137 11 69 277 -139 31 [270-279]
7 281 -141 283 71 -57 -143 287 -1 1 [280-289]
29 291 73 293 -3 -59 -37 33 -149 299 [290-299]
3 301 -151 303 19 -61 -17 307 77 309 [300-309]
31 311 -39 313 -157 -7 79 317 -159 319 [310-319]
-1 321 -161 323 1 13 -163 327 -41 329 [320-329]
33 331 83 37 -167 -67 21 337 -1 339 [330-339]
-17 341 -19 7 -43 -69 -173 347 87 349 [340-349]
-7 39 -11 353 -177 -71 89 357 -179 359 [350-359]
1 | 1 -181 3 91 -73 -183 367 23 41 [360-369]
37 371 93 373 -187 -3 -47 377 -21 379 [370-379]
-19 381 -191 383 -3 -77 -193 43 97 389 [380-389]
39 391 -1 393 -197 -79 11 397 -199 399 [390-399]

Multiplicative.c

2010-02-02T04:39:44Z

Thomas: added HTML tags to make code better displayed on wiki, no changes to code itself

<pre>
/* --- The following code comes from C:\lcc\lib\wizard\textmode.tpl. */
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#define SIZE 200 // how many numbers the computer will look at;
// in principle, increasing this number makes the computer smarter

// our knowledge is encoded in the following array

int vals[SIZE+1]; // vals[i] = +a means f(i) = +f(a); vals[i] = -a means f(i) = -f(a).
int up[SIZE+1]; // best known upper bound for f[1,n]
int low[SIZE+1]; // best known lower bound for f[1,n]

void deduce(void);
void exit_display(void);

void init( void )
{
int i;
int j;

vals[0] = 0;
up[0] = 0;
low[0] = 0;

for (i=1; i <= SIZE; i++)
{
vals[i] = i;
for (j=2; j<40; j++)
{
while (vals[i] % (j*j) == 0) vals[i] /= j*j;
}
if (i % 2 == 0)
{
up[i] = 2; low[i] = -2;
}
else
{
up[i] = 1; low[i] = -1;
}
}
}

int modified=0;

void set( int p, int s ) // set f(p) = s
{
int i;
int pred, sred;

// need to reduce f(p)=s to f(pred)=sred

if (vals[p] > 0)
{
pred = vals[p]; sred = s;
}
else
{
pred = -vals[p]; sred = -s;
}

if (vals[pred] == sred) return;

if (vals[pred] == -sred)
{
printf("Contradiction! f(%d) = 1 and f(%d) = -1 (from f(%d)=%d)\n", pred, pred, p, s);
exit_display();
}

printf("Setting f(%d)=%d...\n",pred,sred);

for (i=1; i<SIZE; i++)
while (vals[i] % pred == 0)
{
vals[i] /= pred; vals[i] *= sred;
}

modified=1;
}

void newup( int i, int new) // add f[1,i] <= new to knowledge
{
if (up[i] <= new) return; // is redundant

if (low[i] > new)
{
printf("Contradiction! %d <= f[1,%d] <= %d\n", low[i], i, new);
exit_display();
}

// printf(" (up[%d] = %d)", i, new);

up[i] = new;
modified = 1;
}

void newdown( int i, int new) // add f[1,i] >= new to knowledge
{
if (low[i] >= new) return; // is redundant

if (up[i] < new)
{
printf("Contradiction! %d >= f[1,%d] >= %d\n", up[i], i, new);
exit_display();
}

// printf(" (low[%d] = %d)", i, new);

low[i] = new;
modified = 1;
}

void cut(int i) // we learn that f[1,i] = 0
{
if (up[i] == 0 && low[i] == 0) return;

if (up[i] < 0)
{
printf("Contradiction! f[1,%d]=-2 but f(%d)=f(%d+1)\n", i, i, i);
exit_display();
}
if (low[i] > 0)
{
printf("Contradiction! f[1,%d]=+2 but f(%d)=f(%d+1)\n", i, i, i);
exit_display();
}

// printf(" (cut[%d])", i);

up[i]=0;
low[i]=0;
modified=1;
}

void deduce( void )
{
int i, j;

do
{
printf("Deducing...\n");
modified = 0;

// begin forward scan to update up,low

for (i=1; i < SIZE; i++)
if (vals[i] == 1 || vals[i] == -1)
{
newup(i, up[i-1]+vals[i]);
newdown(i, low[i-1]+vals[i]);
}

// begin backward scan to update up,low

for (i=SIZE-1; i > 0; i--)
if (vals[i] == 1 || vals[i] == -1)
{
newup(i-1, up[i] - vals[i]);
newdown(i-1, low[i] - vals[i]);
}

// look for cuts

for (i=1; i < SIZE; i++)
if (i % 2 == 1 && vals[i] == vals[i-1])
cut(i-1);

// look for peaks and troughs

for (i=1; i < SIZE-1; i++)
{
if (up[i] <= low[i-1])
set(i,-1);
if (low[i] >= up[i-1])
set(i, 1);
}

// look for multiplicativity opportunities
for (i=2; i <SIZE; i++)
for (j=2; j*i<SIZE; j++)
{
if (vals[i]== -1 || vals[i]==1)
if (vals[j] == -1 || vals[j] == 1)
if (vals[i*j] > 1 || vals[i*j] < -1)
{
printf("f(%d)=f(%d)f(%d)=%d... ", i*j,i,j,vals[i]*vals[j]);
set(i*j, vals[i]*vals[j]);
}

if (vals[i]== -1 || vals[i]==1)
if (vals[i*j] == -1 || vals[i*j] == 1)
if (vals[j] > 1 || vals[j] < -1)
{
printf("f(%d)=f(%d)/f(%d)=%d... ", j,i*j,j,vals[i*j]*vals[i]);
set(j, vals[i]*vals[i*j]);
}
}

}
while(modified);
}

void display( void )
{
int i;

printf("\n");
for (i=0; i < SIZE; i++)
{
if (up[i-1] == 0 && low[i-1] == 0) printf(" |");
else
if (low[i-1] == 2) printf(" ^");
else
if (up[i-1] == -2) printf(" v");
else printf(" ");

printf(" %4d", vals[i]);

if (i % 10 == 9) printf(" [%d-%d]\n", i-9,i);
}

printf("\n");
}

void exit_display(void)
{
display();
exit(1);
}

void main()
{
init();
// set(2, -1);
deduce();
display();
}
</pre>

Short sequences statistics

2010-01-22T18:03:34Z

Thomas: corrected inversion and typo

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Click here to go back to the main experimental results page].

Here are statistics on short sequences obtained by hand (up to length 6), with Alec's code (multiplicative ones), or with Chris's code (the rest). Feel free to add more.

For a discrepancy C=2 there are:
* 2 sequences of length 3
* 10 sequences of length 4
* 14 sequences of length 5
* 26 sequences of length 6
* 44 sequences of length 7
* 88 sequences of length 8
* 100 sequences of length 9
* 152 sequences of length 10
* 240 sequences of length 11
* 370 sequences of length 12
* 556 sequences of length 13
* 882 sequences of length 14
* 750 sequences of length 15
* 1500 sequences of length 16
* 2250 sequences of length 17
* 2784 sequences of length 18
* 4284 sequences of length 19
* 6438 sequences of length 20
* 6062 sequences of length 21
* 9526 sequences of length 22
* 14856 sequences of length 23
* 22944 sequences of length 24
* 26164 sequences of length 25
* 39528 sequences of length 26
* 35122 sequences of length 27
* 54800 sequences of length 28
* 80940 sequences of length 29
* 81326 sequences of length 30
* 122422 sequences of length 31
* 244844 sequences of length 32
* 234934 sequences of length 33
* 356154 sequences of length 34
* 309068 sequences of length 35
* 388042 sequences of length 36
* 589796 sequences of length 37
* 900000 sequences of length 38
* 813466 sequences of length 39
* 1212450 sequences of length 40
* 1837030 sequences of length 41
* 1882194 sequences of length 42
* 2921946 sequences of length 43
* 4544342 sequences of length 44
* 2274560 sequences of length 45
* 3542738 sequences of length 46
* 5495686 sequences of length 47
* 8436986 sequences of length 48, of which 89 are multiplicative
* 9597362 sequences of length 49
* 11352364 sequences of length 50
* 10876536 sequences of length 51
* 16040144 sequences of length 52
* 23626898 sequences of length 53
* 24060696 sequences of length 54
* 22332908 sequences of length 55
* 34665316 sequences of length 56
* 32813078 sequences of length 57
* 48774494 sequences of length 58
* 77333978 sequences of length 59
* 79086932 sequences of length 60
* 118815026 sequences of length 61
* 181723488 sequences of length 62
* 101003862 sequences of length 63
* 202007724 sequences of length 64
* 171112060 sequences of length 65
* 175332604 sequences of length 66
* 266556200 sequences of length 67
* 388397590 sequences of length 68
* 379477372 sequences of length 69
* 355489350 sequences of length 70
* 544231036 sequences of length 71
* 685850834 sequences of length 72
* 1035438526 sequences of length 73
* 1600139990 sequences of length 74
* 967443184 sequences of length 75
* 1435672238 sequences of length 76
* 1335563798 sequences of length 77
* 1305891128 sequences of length 78
* 1985108178 sequences of length 79
* 2984972718 sequences of length 80
* 2245417744 sequences of length 81
* 3449035210 sequences of length 82
* 5401519132 sequences of length 83
* 5272055840 sequences of length 84
* 4470731784 sequences of length 85
* 7013199284 sequences of length 86
* 6544305958 sequences of length 87
* 10138992314 sequences of length 88
* 15284191798 sequences of length 89
* 9839916650 sequences of length 90
* 8579138578 sequences of length 91
* 13353765596 sequences of length 92
* 12170187396 sequences of length 93
* 18839898968 sequences of length 94
* 17144676512 sequences of length 95
* 26322363104 sequences of length 96, of which 119 are multiplicative
* 39457407576 sequences of length 97
* 48303722254 sequences of length 98
* 27179767280 sequences of length 99
* 31055060100 sequences of length 100
* ...
* ????? ? sequences of length 192, of which 304 are multiplicative

More precise data about multiplicave sequences themselves is available [http://michaelnielsen.org/polymath1/index.php?title=Multiplicative_sequences on this page].

Experimental results

2010-01-21T09:43:46Z

Thomas: Removed incorrect 3250 sequence

[[The Erdős discrepancy problem|To return to the main Polymath5 page, click here]].

Perhaps we should have two kinds of subpages to this page: Pages about finding examples, and pages about analyzing them?

== Experimental data==
* [[The first 1124-sequence]] with discrepancy 2. ''Some more description''
* Other [[length 1124 sequences]] with discrepancy 2. ''Some more description''
* A [[sequence of length 1091]] with discrepancy 2.
* A [[sequence of length 1112]] derived from one with nice multiplicative properties.
* [[Sequences given by modulated Sturmian functions]].
* Some data about the problem with [[different upper and lower bound]]. Let N(a,b) be the largest N such that there is a sequence <math>x_1,\dots,x_N</math> all of whose HAP-errors are between -a and b, inclusive.
* Sequences taking values in <math>\mathbb{T}</math>:
** [[4th roots of unity]]
** [[6th roots of unity]]
* [http://thomas1111.wordpress.com/2010/01/10/tables-for-a-c10-candidate/ A sequence of length 407] with discrepancy 2 such that <math>x_n=x_{32 n}</math> for every n. [[The HAP-subsequence structure of that sequence]].
* More [[T32-invariant sequences]].
* Long [[multiplicative sequences]].
* Long sequences satisfying [[T2(x) = -x]].
* Long sequences satisfying [[T2(x) = T5(x) = -x]]
* Long sequences satisfying [[T2(x) = -T3(x)]].
* Long sequences satisfying constraints of the form [[T_m(x) = (+/-)T_n(x)]].
* Table of [[longest constrained sequences]].
* Table of [[short sequences statistics]].
* [[Dirichlet inverses]] of good sequences.
* Sequences with [[bounded Dirichlet inverse]].
* [[Shifts and signs]] related to an interesting structure of the first 1124-sequence.

==Source code==

* [[Convert raw input string into CSV table]]
* [[Create tables in an HTML file from an input sequence]]
* [[Verify the bounded discrepancy of an input sequence]]
* [[Depth-first search]]
* [[Search for completely multiplicative sequences]]
* [[Refined greedy computation of multiplicative sequences]]
* [[Computing a HAP basis]]
* [[Estimate the number of discrepancy 2 sequences]]

==Wish list==

There is a separate page for [[proposals for finding long low-discrepancy sequences]]. It goes without saying that implementing any of these proposals belongs to the wish list.


* Find long/longest quasi-multiplicative sequences with some fixed group G, function <math>G\to \{-1,1\}</math> and maximal discrepancy C
** <math>G=C_6</math> and the function that sends 0,1 and 2 to 1 (because this seems to be a good choice)
* Do a "Mark-Bennet-style analysis" of one of the new 1124-sequences. [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4827] Also [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4842 done] (by Mark Bennet).
*. Take a moderately large k and search for the longest sequence of discrepancy 2 that's constructed as follows. First, pick a completely multiplicative function f to the group <math>C_{2k}</math>. Then set <math>x_n</math> to be 1 if f(n) lies between 0 and k-1, and -1 if f(n) lies between k and 2k-1. Alec has already [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4563 done this for k=1] and [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4734 partially done it for k=3].
*Search for the longest sequence of discrepancy 2 with the property that <math>x_n=x_{32n}</math> for every n. The motivation for this is to produce a fundamentally different class of examples (different because their group structure would include an element of order 5). It's not clear that it will work, since 32 is a fairly large number. However, if you've chosen <math>x_{32n}</math> then that will have some influence on several other choices, such as <math>x_{4n},x_{8n}</math> and <math>x_{16n}</math>, so maybe it will lead to something interesting. Alec [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4873 has made a start on this] and an [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4874 initial investigation] suggests that the sequence he has found does indeed have some <math>C_{10}</math>-related structure.
*Here's another experiment that should be pretty easy to program and might yield something interesting. It's to look at the how the discrepancy appears to grow when you define a sequence using a greedy algorithm. I say "a" greedy algorithm because there are various algorithms that could reasonably be described as greedy. Here are a few.

<blockquote>
1. For each n let <math>x_n</math> be chosen so as to minimize the discrepancy so far, given the choices already made for <math>x_1,\dots,x_{n-1}</math>. (If this does not uniquely determine <math>x_n</math> then choose it arbitrarily, or randomly, or according to some simple rule like always equalling 1.)
</blockquote>

<blockquote>
2. Same as 1 but with additional constraints, in the hope that these make the sequence more likely to be good. For instance, one might insist that <math>x_{2k}=x_{3k}</math> for every k. Here, when choosing <math>x_n</math> one would probably want to minimize the discrepancy up to <math>x_{n+k}</math> if <math>x_{n+1},\dots,x_{n+k}</math> had already been chosen. Another obvious constraint to try is complete multiplicativity.
</blockquote>

<blockquote>
3. A greedy algorithm of sorts, but this time trying to minimize a different parameter. The first algorithm will do this: when you pick <math>x_n</math> you look, for each factor d of n, at the partial sum along the multiples of d up to but not including n. This will give you a set A of numbers (the possible partial sums). If max(A) is greater than max(-A) then you set <math>x_n=-1</math>, if max(-A) is greater than max(A) then you let <math>x_n=1</math>, and if they are equal then you make the decision according to some rule that seems sensible. But it might be that you would end up with a slower-growing discrepancy if you regarded A as a multiset and made the decision on some other basis. For instance, you could take the sum of <math>2^k</math> over all positive elements <math>k\in A</math> (with multiplicity) and the sum of <math>2^{-k}</math> over all negative elements and choose <math>x_n</math> according to which was bigger. Although that wouldn't minimize the discrepancy at each stage, it might make the sequence better for future development because it wouldn't sacrifice the needs of an overwhelming majority to those of a few rogue elements.
</blockquote>

<blockquote>
4. A greedy algorithm to choose a good completely multiplicative low-discrepancy sequence. Now you are free only to choose the values at primes. If you have chosen the values up to but not including p, then fill in all the values that are forced by multiplicativity and then make whatever seems to be the best choice for the value at p. Again, there are several approaches that could be reasonable here. One is simply to ensure that the partial sum of the sequence up to p is as small (in modulus) as you can make it. But that would be foolish if you've already filled in the values at p+1,...,p+k. So an only slightly less greedy algorithm is to look at the effect of your choice at p on the partial sums all the way up to the next prime and choose the best value accordingly. If you do that, then at what rate do the partial sums grow? In particular, do they grow at least logarithmically? [http://michaelnielsen.org/polymath1/index.php?title=Multiplicative_sequences#Minimizing_D_up_to_the_next_prime This is being addressed here]
</blockquote>

<blockquote>
The motivation for these experiments is to see whether they, or some variants, appear to lead to sublogarithmic growth. If they do, then we could start trying to prove rigorously that sublogarithmic growth is possible. I still think that a function that arises in nature and satisfies f(1124)=2 ought to be sublogarithmic.
</blockquote>

*What happens if one applies a backtracking algorithm to try to extend the following discrepancy-2 sequence, which satisfies <math>x_{2n}=-x_n</math> for every n, to a much longer discrepancy-2 sequence: + - - + - + + - - + + - + - + + - + - - + - - + + - - + + - + - - + - - + + + + - - - + + + - - + - + + - + - - + - ? This question has been answered [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4893 in the comments following the asking of the question on the blog].

* Investigate what happens if our HAPs are restricted to allow differences divisible only by 2 or 3 [and then other sets of primes including 2] - {2,3,5,7} would be interesting - is there an infinite sequence of discrepancy 2 in these simple cases - is it easy to find an infinite sequence with finite discrepancy in these cases? [for sets of odd primes, take a sequence which is 1 on odd numbers, -1 on even numbers. Including 2 is the non-trivial case]. It is possible that completely multiplicative sequences could be found for some of these cases.

* Compute the Dirichlet series <math>f(s) = \sum x_n n^{-s}</math> for some of our long low-discrepancy series, and see what this function looks like in the vicinity of <math>1</math>, and elsewhere. [http://gowers.wordpress.com/2010/01/11/the-erds-discrepancy-problem-iii/#comment-5062 Alec has now looked at this].

*Take a long sequence <math>(x_n)</math> of discrepancy 2 and try to create a new long sequence <math>(y_n)</math> subject to the constraint that <math>y_{2n}=x_n</math>. How far does one typically get before getting stuck? And how much further does one get if one uses the resulting sequence as a seed for the usual algorithm? [http://gowers.wordpress.com/2010/01/14/the-erds-discrepancy-problem-iv/#comment-5096 One does not get too far, as Alec showed].

*Take some of the good sequences and calculate the following parameter, which is supposed to measure the amount of multiplicity. If the sequence is defined up to n, then the parameter is the expected value of <math>x_ax_bx_cx_d</math> over all quadruples (a,b,c,d) such that a,b,c,d are at most n and ab=cd. I'm expecting that the answer will be significantly greater than zero -- perhaps something like 0.3 -- but would like to have this confirmed. [http://gowers.wordpress.com/2010/01/14/the-erds-discrepancy-problem-iv/#comment-5132 It has now been confirmed].

* ... you are welcome to add more.

Sequence of length 3250

2010-01-21T09:42:43Z

Thomas: Removed

Incorrect item removed, see discussion page and previous versions of the page.

Talk:Sequence of length 3250

2010-01-21T09:41:52Z

Thomas: Point out that this sequence doesn't have discrepancy 2

Dear Mathbouchard,
unfortunately this sequence doesn't have discrepancy 2, as can be seen from its beginning - - - + + - - +... which immediately hits 3 (that's -1, -2, -3). The full sequence in fact has discrepancy 58.
-Thomas.

Refined greedy computation of multiplicative sequences

2010-01-15T14:15:17Z

Thomas: corrected silly typo in function lharmw

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Back to the main experimental results page].

Here is the source of a C++ program for investigating refined greedy algorithms in the case of completely-multiplicative sequences.

The freedom lies in choosing the values at prime indices. Many methods can be devised but only a few are implemented for the moment, feel free to add many more (please indicate what you've added or modified). The code is commented and hopefully fairly straightforward (it probably could be optimized here and there, but has been carefully checked and seems correct).

The input file must be in the format of a sequence separated by spaces like + - + (with a space after the last sign, no carriage returns). These indicate the values at the first primes: <math>x_2</math>, <math>x_3</math>, <math>x_5</math>... In fact it is also possible not to specify any one value before the end by putting a 0.

There are three output files: the first one will contain the sequence found, the second may contain information on the choices made for each undetermined prime (I've commented it out for now), and the third contains in columns: n, D(n), partial sum of sequence up to n, partial sum of 2-HAP up to n, ... 11-HAP up to n (you can add more if so you wish).

Finally the constant M is the length of the sequence to be computed.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
13-jan-2010: written from scratch. Really basic code...
14-jan-2010: implemented various methods to choose values
at prime indices.
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------

int const M=3000;

int tab[M], primes[M], prtab[M], glopartsum[M];
char filenameIN[50], filenameOUT[50], filenameOUTT[50], filenameOUTTT[50];
ofstream myfileOUT;//sequence
ofstream myfileOUTT;//other data
ofstream myfileOUTTT;//yet other data
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//---------
//mypow
//integer power: a^b
//---------
int mypow(int a, int b){
int i, ans=1;
for(i=1;i<=b;i++){
ans*=a;
}
if(b==0){
return 1;
}
else{
return ans;
}
}

//---------
//primfact
//compute prime factors and store
//returns 2 iff n is prime
//---------
int primfact(int n){

int a, b, c, d, ncurr;
for(a=0;a<M;a++){
prtab[a]=0;
}
ncurr=n;
for(b=0;b<n;b++){//surely the n-th prime is greater than n already
while(primes[b]!=0 && ncurr>1 && (ncurr%primes[b])==0){
prtab[b]+=1;
ncurr/=primes[b];
}

if(primes[b]==0){
//exit
b+=M;
}
}
d=0;
for(c=0;c<M;c++){
if(prtab[c]!=0){
d+=prtab[c];
}
}
if(d==1){
//cout<<" "<<n<<" is prime"<<endl;
return 2;
}
if(d!=1){
//cout<<" "<<n<<" is composite"<<endl;
return 0;
}
}

//---------
//disc
//computes discrepancy of sequence of +/-1 contained in tabb
//over all HAPs up to len
//store signed partial sums up to len:
// glopartsum[0]=0, glopartsum[1]=sum of seq, glopartsum[2]=sum of 2-HAP...
//---------
int disc(int tabb[M], int len){

int n, d, i;
int signsum;
int absum;
int r;
int beflocdisc;
int aflocdisc;

beflocdisc=0;
aflocdisc=0;

for(n=2;n<=len;n++){
for(d=1;d<n;d++){
signsum=0;
absum=0;
glopartsum[d]=0;
r=int(floor(n/d));
for(i=1;i<=r;i++){
signsum+=tabb[d*i-1];
}
glopartsum[d]=signsum;
absum=abs(signsum);
aflocdisc=max(beflocdisc,absum);
beflocdisc=aflocdisc;
}
}
return aflocdisc;
}

//---------
//lplain
//just adds the entries of tabb
//---------
int lplain(int tabb[M], int len){
int norm, a;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[a];
}
return norm;
}

//---------
//l1norm
//---------
int l1norm(int tabb[M], int len){
int a, norm;
norm=0;
for(a=0;a<len;a++){
norm+=abs(tabb[a]);
}
return norm;
}

//---------
//l2norm
//---------
double l2norm(int tabb[M], int len){
int a, snorm;
double norm;
snorm=0; norm=0.0;
for(a=0;a<len;a++){
snorm+=(tabb[a]*tabb[a]);
}
norm=sqrt(snorm);
return norm;
}

//---------
//lharmw
//harmonic weighting
//---------
double lharmw(int tabb[M], int len){
int a;
double norm;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[a]/(a+1);
}
return norm;
}

//----------------------------------------------------------
//-- main:
//
//we compute a multiplicative sequence and its discrepancy by
//specifying its values at first p primes
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, w, j, c, t, imax=0, maxlength, currprime, iprec;
int datab[M];
int mycontinue, g, loclen, h;
double nmplus, nmminus;
string line, buff;
string myplus ("+");
string myminus ("-");
string myzero ("0");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output LONG MULT SEQUENCE filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

cout<<endl;
cout<<" give output CHOICE filename:"<<endl;
cin>>filenameOUTT;
cout<<endl;

cout<<endl;
cout<<" give output DISC & PARTSUM filename: "<<endl;
cin>>filenameOUTTT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

myfileOUTT.open(filenameOUTT,ios::out);
myfileOUTT.close();

myfileOUTTT.open(filenameOUTTT,ios::out);
myfileOUTTT.close();

//fill tables with zeros
for(i=0;i<M;i++){
tab[i];
datab[i]=0;
primes[i]=0;
prtab[i]=0;
}

//construct and store primes up to M
//primes[a] = (a+1)-th prime
iprec=2;
for(i=2;i<=M;i++){
k=0;
for(j=1;j<=i;j++){
if( (i%j)==0){
k++;
}
}
if(k==2){
primes[iprec-2]=i;
iprec+=1;
}
}

//-- read the choices of values at first few primes
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;
getline(myfileIN,line);
stringstream stsm(line);
while(stsm>>buff){
currprime=primes[i];
if(buff.compare(myplus)==0){
datab[currprime-1]=1;
}
if(buff.compare(myminus)==0){
datab[currprime-1]=-1;
}
if(buff.compare(myzero)==0){
datab[currprime-1]=0;
}
if( datab[currprime-1] !=1 && datab[currprime-1] !=-1 && datab[currprime-1]!=0){
cout<<"problem: element number "<<i+1<<" is neither + nor - nor 0."<<endl;
abort();
}
i+=1;
}
imax=i;
cout<<" this initial sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}
myfileIN.close();

//-- main processing:
//fill the rest of the sequence multiplicatively
//and according to chosen method

//convention: datab[i]=x_{i+1}
//and first index is always associated to +
datab[0]=1;

mycontinue=0;

while(mycontinue==0){

for(u=2;u<M;u++){

if(datab[u-1]==0){

k=0;
k=primfact(u);

//if prime: there's freedom, choose one method to fill it
if(k==2){

//choice 1: silly uniform -1
//datab[u-1]=-1;

//choice 2: take into account of partial sums
loclen=u;

cout<<" adressing "<<loclen<<endl;

//values are known below loclen, not at prime loclen,
//and from loclen+1 to loclen+h. we then choose loclen
//(lots of flexibility)

//compute h
h=1;
for(v=loclen+1;v<M;v++){
if(datab[v]==0){
h=v-loclen;
v+=M;
}
}
cout<<" values after are known up to "<<loclen+h<<endl;

//now compute the effect of imposing datab[u-1]=+1
//would have on partial sums up to loclen+h-1
//you can change lplain to l2norm or whatever other idea
datab[u-1]=+1;
c=disc(datab,loclen+h-1);
nmplus=lplain(datab,loclen+h-1);

//compute the same with -1 instead
datab[u-1]=-1;
c=disc(datab,loclen+h-1);
nmminus=lplain(datab,loclen+h-1);

//now choose
if(abs(nmplus)>abs(nmminus)){
datab[u-1]=-1;
}
else{
datab[u-1]=+1;
}
cout<<" nmplus="<<nmplus<<", nmminus="<<nmminus<<endl;

/*------temporarily removed, could be useful to monitor
//record those numbers for further analysis
myfileOUTT.open(filenameOUTT,ios::app);
myfileOUTT<<u<<"\t"<<nmplus<<"\t"<<nmminus<<endl;
myfileOUTT.close();
------------------------*/

//choice 3:
//more ideas to be added...

//in any case: give info
cout<<" imposing: x_"<<u<<"="<<datab[u-1]<<endl;

//------------ in theory there's nothing to change beyond this point, except
// possibily the number of HAP's partial sums printed in the third file

}
else{//if composite then try to compute it multiplicatively

//loop on prime factors of u to see what is known so far

w=1;
for(j=0;j<M;j++){

if(prtab[j]!=0 && datab[ primes[j]-1 ]!=0){
//then this x_j is known
w*=1;
}

if(prtab[j]!=0 && datab[ primes[j]-1 ]==0){
//then this x_j not known, enough to discard
w*=0;
}
}

//if all x_j are known then compute x_u
if(w==1){
datab[u-1]=1;
for(j=0;j<M;j++){
if(prtab[j]!=0){
datab[u-1]*=mypow(datab[ primes[j]-1 ],prtab[j]);
}
}
cout<<" just computed x_"<<u<<"="<<datab[u-1]<<endl;
}
else{
//cout<<" cannot yet compute x_"<<u<<endl;
//cout<<endl;
}

}//end of mult comp attempt
}//end of this u
else{
//cout<<" already known"<<endl;
}
}//end of u loop

//now see whether we should go on
mycontinue=1;
for(c=0;c<M-1;c++){
if(datab[c]!=0){
mycontinue*=1;
}
else{
mycontinue*=0;
}
}

}//end of mycontinue

cout<<" ---now out, computing partial sums"<<endl;

myfileOUT.open(filenameOUT,ios::app);
myfileOUTTT.open(filenameOUTTT,ios::app);
for(u=0;u<M;u++){

if(u%50==0){
cout<<" computed "<<u<<" data points"<<endl;
}

//first output discrepancy and partial sums up to length u, and 11-th HAP
k=disc(datab,u);
if(u>0){
myfileOUTTT<<u<<"\t"<<k<<"\t";
for(d=1;d<u;d++){
if(d<=11){
myfileOUTTT<<glopartsum[d]<<"\t";
}
}
myfileOUTTT<<"\n";
}

//now output sequence value
if(datab[u]==1){
myfileOUT<<"+ ";
}
if(datab[u]==-1){
myfileOUT<<"- ";
}
}
myfileOUT.close();
myfileOUTTT.close();

cout<<" discrepancy="<<disc(datab,M)<<endl;

//exit normally
return 0;
}
</pre>

Multiplicative sequences

2010-01-15T13:56:09Z

Thomas:

Multiplicative sequences

2010-01-15T13:42:12Z

Thomas:

Multiplicative sequences

2010-01-15T13:41:02Z

Thomas:

Experimental results

2010-01-15T13:35:02Z

Thomas:

[[The Erdős discrepancy problem|To return to the main Polymath5 page, click here]].

Perhaps we should have two kinds of subpages to this page: Pages about finding examples, and pages about analyzing them?

== Experimental data==
* [[The first 1124-sequence]] with discrepancy 2. ''Some more description''
* Other [[length 1124 sequences]] with discrepancy 2. ''Some more description''
* A [[sequence of length 1112]] derived from one with nice multiplicative properties.
* Some data about the problem with [[different upper and lower bound]]. Let N(a,b) be the largest N such that there is a sequence <math>x_1,\dots,x_N</math> all of whose HAP-errors are between -a and b, inclusive.
* Sequences taking values in <math>\mathbb{T}</math>:
** [[4th roots of unity]]
** [[6th roots of unity]]
* [http://thomas1111.wordpress.com/2010/01/10/tables-for-a-c10-candidate/ A sequence of length 407] with discrepancy 2 such that <math>x_n=x_{32 n}</math> for every n. [[The HAP-subsequence structure of that sequence]].
* More [[T32-invariant sequences]].
* Long [[multiplicative sequences]].
* Long sequences satisfying [[T2(x) = -x]].
* Long sequences satisfying [[T2(x) = T5(x) = -x]]
* Long sequences satisfying [[T2(x) = -T3(x)]].
* Long sequences satisfying constraints of the form [[T_m(x) = (+/-)T_n(x)]].
* Table of [[longest constrained sequences]].
* Table of [[short sequences statistics]].
* [[Dirichlet inverses]] of good sequences.

==Source code==

* [[Convert raw input string into CSV table]]
* [[Create tables in an HTML file from an input sequence]]
* [[Verify the bounded discrepancy of an input sequence]]
* [[Depth-first search]]
* [[Search for completely multiplicative sequences]]
* [[Refined greedy computation of multiplicative sequences]]
* [[Computing a HAP basis]]

==Wish list==

There is a separate page for [[proposals for finding long low-discrepancy sequences]]. It goes without saying that implementing any of these proposals belongs to the wish list.


* Find long/longest quasi-multiplicative sequences with some fixed group G, function <math>G\to \{-1,1\}</math> and maximal discrepancy C
** <math>G=C_6</math> and the function that sends 0,1 and 2 to 1 (because this seems to be a good choice)
* Do a "Mark-Bennet-style analysis" of one of the new 1124-sequences. [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4827] Also [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4842 done] (by Mark Bennet).
*. Take a moderately large k and search for the longest sequence of discrepancy 2 that's constructed as follows. First, pick a completely multiplicative function f to the group <math>C_{2k}</math>. Then set <math>x_n</math> to be 1 if f(n) lies between 0 and k-1, and -1 if f(n) lies between k and 2k-1. Alec has already [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4563 done this for k=1] and [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4734 partially done it for k=3].
*Search for the longest sequence of discrepancy 2 with the property that <math>x_n=x_{32n}</math> for every n. The motivation for this is to produce a fundamentally different class of examples (different because their group structure would include an element of order 5). It's not clear that it will work, since 32 is a fairly large number. However, if you've chosen <math>x_{32n}</math> then that will have some influence on several other choices, such as <math>x_{4n},x_{8n}</math> and <math>x_{16n}</math>, so maybe it will lead to something interesting. Alec [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4873 has made a start on this] and an [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4874 initial investigation] suggests that the sequence he has found does indeed have some <math>C_{10}</math>-related structure.
*Here's another experiment that should be pretty easy to program and might yield something interesting. It's to look at the how the discrepancy appears to grow when you define a sequence using a greedy algorithm. I say "a" greedy algorithm because there are various algorithms that could reasonably be described as greedy. Here are a few.

<blockquote>
1. For each n let <math>x_n</math> be chosen so as to minimize the discrepancy so far, given the choices already made for <math>x_1,\dots,x_{n-1}</math>. (If this does not uniquely determine <math>x_n</math> then choose it arbitrarily, or randomly, or according to some simple rule like always equalling 1.)
</blockquote>

<blockquote>
2. Same as 1 but with additional constraints, in the hope that these make the sequence more likely to be good. For instance, one might insist that <math>x_{2k}=x_{3k}</math> for every k. Here, when choosing <math>x_n</math> one would probably want to minimize the discrepancy up to <math>x_{n+k}</math> if <math>x_{n+1},\dots,x_{n+k}</math> had already been chosen. Another obvious constraint to try is complete multiplicativity.
</blockquote>

<blockquote>
3. A greedy algorithm of sorts, but this time trying to minimize a different parameter. The first algorithm will do this: when you pick <math>x_n</math> you look, for each factor d of n, at the partial sum along the multiples of d up to but not including n. This will give you a set A of numbers (the possible partial sums). If max(A) is greater than max(-A) then you set <math>x_n=-1</math>, if max(-A) is greater than max(A) then you let <math>x_n=1</math>, and if they are equal then you make the decision according to some rule that seems sensible. But it might be that you would end up with a slower-growing discrepancy if you regarded A as a multiset and made the decision on some other basis. For instance, you could take the sum of <math>2^k</math> over all positive elements <math>k\in A</math> (with multiplicity) and the sum of <math>2^{-k}</math> over all negative elements and choose <math>x_n</math> according to which was bigger. Although that wouldn't minimize the discrepancy at each stage, it might make the sequence better for future development because it wouldn't sacrifice the needs of an overwhelming majority to those of a few rogue elements.
</blockquote>

<blockquote>
4. A greedy algorithm to choose a good completely multiplicative low-discrepancy sequence. Now you are free only to choose the values at primes. If you have chosen the values up to but not including p, then fill in all the values that are forced by multiplicativity and then make whatever seems to be the best choice for the value at p. Again, there are several approaches that could be reasonable here. One is simply to ensure that the partial sum of the sequence up to p is as small (in modulus) as you can make it. But that would be foolish if you've already filled in the values at p+1,...,p+k. So an only slightly less greedy algorithm is to look at the effect of your choice at p on the partial sums all the way up to the next prime and choose the best value accordingly. If you do that, then at what rate do the partial sums grow? In particular, do they grow at least logarithmically? [http://michaelnielsen.org/polymath1/index.php?title=Multiplicative_sequences#Minimizing_D_up_to_the_next_prime This is being adressed here]
</blockquote>

<blockquote>
The motivation for these experiments is to see whether they, or some variants, appear to lead to sublogarithmic growth. If they do, then we could start trying to prove rigorously that sublogarithmic growth is possible. I still think that a function that arises in nature and satisfies f(1124)=2 ought to be sublogarithmic.
</blockquote>

*What happens if one applies a backtracking algorithm to try to extend the following discrepancy-2 sequence, which satisfies <math>x_{2n}=-x_n</math> for every n, to a much longer discrepancy-2 sequence: + - - + - + + - - + + - + - + + - + - - + - - + + - - + + - + - - + - - + + + + - - - + + + - - + - + + - + - - + - ? This question has been answered [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4893 in the comments following the asking of the question on the blog].

* Investigate what happens if our HAPs are restricted to allow differences divisible only by 2 or 3 [and then other sets of primes including 2] - {2,3,5,7} would be interesting - is there an infinite sequence of discrepancy 2 in these simple cases - is it easy to find an infinite sequence with finite discrepancy in these cases? [for sets of odd primes, take a sequence which is 1 on odd numbers, -1 on even numbers. Including 2 is the non-trivial case]. It is possible that completely multiplicative sequences could be found for some of these cases.

* Compute the Dirichlet series <math>f(s) = \sum x_n n^{-s}</math> for some of our long low-discrepancy series, and see what this function looks like in the vicinity of <math>1</math>, and elsewhere. [http://gowers.wordpress.com/2010/01/11/the-erds-discrepancy-problem-iii/#comment-5062 Alec has now looked at this].

*Take a long sequence <math>(x_n)</math> of discrepancy 2 and try to create a new long sequence <math>(y_n)</math> subject to the constraint that <math>y_{2n}=x_n</math>. How far does one typically get before getting stuck? And how much further does one get if one uses the resulting sequence as a seed for the usual algorithm?

* ... you are welcome to add more.

Multiplicative sequences

2010-01-15T13:31:42Z

Thomas: added data on minimizing D

Refined greedy computation of multiplicative sequences

2010-01-14T21:29:54Z

Thomas: clarified many comments within the code!

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Back to the main experimental results page].

Here is the source of a C++ program for investigating refined greedy algorithms in the case of completely-multiplicative sequences.

The freedom lies in choosing the values at prime indices. Many methods can be devised but only a few are implemented for the moment, feel free to add many more (please indicate what you've added or modified). The code is commented and hopefully fairly straightforward (it probably could be optimized here and there, but has been carefully checked and seems correct).

The input file must be in the format of a sequence separated by spaces like + - + (with a space after the last sign, no carriage returns). These indicate the values at the first primes: <math>x_2</math>, <math>x_3</math>, <math>x_5</math>... In fact it is also possible not to specify any one value before the end by putting a 0.

There are three output files: the first one will contain the sequence found, the second may contain information on the choices made for each undetermined prime (I've commented it out for now), and the third contains in columns: n, D(n), partial sum of sequence up to n, partial sum of 2-HAP up to n, ... 11-HAP up to n (you can add more if so you wish).

Finally the constant M is the length of the sequence to be computed.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
13-jan-2010: written from scratch. Really basic code...
14-jan-2010: implemented various methods to choose values
at prime indices.
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------

int const M=3000;

int tab[M], primes[M], prtab[M], glopartsum[M];
char filenameIN[50], filenameOUT[50], filenameOUTT[50], filenameOUTTT[50];
ofstream myfileOUT;//sequence
ofstream myfileOUTT;//other data
ofstream myfileOUTTT;//yet other data
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//---------
//mypow
//integer power: a^b
//---------
int mypow(int a, int b){
int i, ans=1;
for(i=1;i<=b;i++){
ans*=a;
}
if(b==0){
return 1;
}
else{
return ans;
}
}

//---------
//primfact
//compute prime factors and store
//returns 2 iff n is prime
//---------
int primfact(int n){

int a, b, c, d, ncurr;
for(a=0;a<M;a++){
prtab[a]=0;
}
ncurr=n;
for(b=0;b<n;b++){//surely the n-th prime is greater than n already
while(primes[b]!=0 && ncurr>1 && (ncurr%primes[b])==0){
prtab[b]+=1;
ncurr/=primes[b];
}

if(primes[b]==0){
//exit
b+=M;
}
}
d=0;
for(c=0;c<M;c++){
if(prtab[c]!=0){
d+=prtab[c];
}
}
if(d==1){
//cout<<" "<<n<<" is prime"<<endl;
return 2;
}
if(d!=1){
//cout<<" "<<n<<" is composite"<<endl;
return 0;
}
}

//---------
//disc
//computes discrepancy of sequence of +/-1 contained in tabb
//over all HAPs up to len
//store signed partial sums up to len:
// glopartsum[0]=0, glopartsum[1]=sum of seq, glopartsum[2]=sum of 2-HAP...
//---------
int disc(int tabb[M], int len){

int n, d, i;
int signsum;
int absum;
int r;
int beflocdisc;
int aflocdisc;

beflocdisc=0;
aflocdisc=0;

for(n=2;n<=len;n++){
for(d=1;d<n;d++){
signsum=0;
absum=0;
glopartsum[d]=0;
r=int(floor(n/d));
for(i=1;i<=r;i++){
signsum+=tabb[d*i-1];
}
glopartsum[d]=signsum;
absum=abs(signsum);
aflocdisc=max(beflocdisc,absum);
beflocdisc=aflocdisc;
}
}
return aflocdisc;
}

//---------
//lplain
//just adds the entries of tabb
//---------
int lplain(int tabb[M], int len){
int norm, a;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[a];
}
return norm;
}

//---------
//l1norm
//---------
int l1norm(int tabb[M], int len){
int a, norm;
norm=0;
for(a=0;a<len;a++){
norm+=abs(tabb[a]);
}
return norm;
}

//---------
//l2norm
//---------
double l2norm(int tabb[M], int len){
int a, snorm;
double norm;
snorm=0; norm=0.0;
for(a=0;a<len;a++){
snorm+=(tabb[a]*tabb[a]);
}
norm=sqrt(snorm);
return norm;
}

//---------
//lharmw
//harmonic weighting
//---------
double lharmw(int tabb[M], int len){
int a;
double norm;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[M]/(a+1);
}
return norm;
}

//----------------------------------------------------------
//-- main:
//
//we compute a multiplicative sequence and its discrepancy by
//specifying its values at first p primes
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, w, j, c, t, imax=0, maxlength, currprime, iprec;
int datab[M];
int mycontinue, g, loclen, h;
double nmplus, nmminus;
string line, buff;
string myplus ("+");
string myminus ("-");
string myzero ("0");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output LONG MULT SEQUENCE filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

cout<<endl;
cout<<" give output CHOICE filename:"<<endl;
cin>>filenameOUTT;
cout<<endl;

cout<<endl;
cout<<" give output DISC & PARTSUM filename: "<<endl;
cin>>filenameOUTTT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

myfileOUTT.open(filenameOUTT,ios::out);
myfileOUTT.close();

myfileOUTTT.open(filenameOUTTT,ios::out);
myfileOUTTT.close();

//fill tables with zeros
for(i=0;i<M;i++){
tab[i];
datab[i]=0;
primes[i]=0;
prtab[i]=0;
}

//construct and store primes up to M
//primes[a] = (a+1)-th prime
iprec=2;
for(i=2;i<=M;i++){
k=0;
for(j=1;j<=i;j++){
if( (i%j)==0){
k++;
}
}
if(k==2){
primes[iprec-2]=i;
iprec+=1;
}
}

//-- read the choices of values at first few primes
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;
getline(myfileIN,line);
stringstream stsm(line);
while(stsm>>buff){
currprime=primes[i];
if(buff.compare(myplus)==0){
datab[currprime-1]=1;
}
if(buff.compare(myminus)==0){
datab[currprime-1]=-1;
}
if(buff.compare(myzero)==0){
datab[currprime-1]=0;
}
if( datab[currprime-1] !=1 && datab[currprime-1] !=-1 && datab[currprime-1]!=0){
cout<<"problem: element number "<<i+1<<" is neither + nor - nor 0."<<endl;
abort();
}
i+=1;
}
imax=i;
cout<<" this initial sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}
myfileIN.close();

//-- main processing:
//fill the rest of the sequence multiplicatively
//and according to chosen method

//convention: datab[i]=x_{i+1}
//and first index is always associated to +
datab[0]=1;

mycontinue=0;

while(mycontinue==0){

for(u=2;u<M;u++){

if(datab[u-1]==0){

k=0;
k=primfact(u);

//if prime: there's freedom, choose one method to fill it
if(k==2){

//choice 1: silly uniform -1
//datab[u-1]=-1;

//choice 2: take into account of partial sums
loclen=u;

cout<<" adressing "<<loclen<<endl;

//values are known below loclen, not at prime loclen,
//and from loclen+1 to loclen+h. we then choose loclen
//(lots of flexibility)

//compute h
h=1;
for(v=loclen+1;v<M;v++){
if(datab[v]==0){
h=v-loclen;
v+=M;
}
}
cout<<" values after are known up to "<<loclen+h<<endl;

//now compute the effect of imposing datab[u-1]=+1
//would have on partial sums up to loclen+h-1
//you can change lplain to l2norm or whatever other idea
datab[u-1]=+1;
c=disc(datab,loclen+h-1);
nmplus=lplain(datab,loclen+h-1);

//compute the same with -1 instead
datab[u-1]=-1;
c=disc(datab,loclen+h-1);
nmminus=lplain(datab,loclen+h-1);

//now choose
if(abs(nmplus)>abs(nmminus)){
datab[u-1]=-1;
}
else{
datab[u-1]=+1;
}
cout<<" nmplus="<<nmplus<<", nmminus="<<nmminus<<endl;

/*------temporarily removed, could be useful to monitor
//record those numbers for further analysis
myfileOUTT.open(filenameOUTT,ios::app);
myfileOUTT<<u<<"\t"<<nmplus<<"\t"<<nmminus<<endl;
myfileOUTT.close();
------------------------*/

//choice 3:
//more ideas to be added...

//in any case: give info
cout<<" imposing: x_"<<u<<"="<<datab[u-1]<<endl;

//------------ in theory there's nothing to change beyond this point, except
// possibily the number of HAP's partial sums printed in the third file

}
else{//if composite then try to compute it multiplicatively

//loop on prime factors of u to see what is known so far

w=1;
for(j=0;j<M;j++){

if(prtab[j]!=0 && datab[ primes[j]-1 ]!=0){
//then this x_j is known
w*=1;
}

if(prtab[j]!=0 && datab[ primes[j]-1 ]==0){
//then this x_j not known, enough to discard
w*=0;
}
}

//if all x_j are known then compute x_u
if(w==1){
datab[u-1]=1;
for(j=0;j<M;j++){
if(prtab[j]!=0){
datab[u-1]*=mypow(datab[ primes[j]-1 ],prtab[j]);
}
}
cout<<" just computed x_"<<u<<"="<<datab[u-1]<<endl;
}
else{
//cout<<" cannot yet compute x_"<<u<<endl;
//cout<<endl;
}

}//end of mult comp attempt
}//end of this u
else{
//cout<<" already known"<<endl;
}
}//end of u loop

//now see whether we should go on
mycontinue=1;
for(c=0;c<M-1;c++){
if(datab[c]!=0){
mycontinue*=1;
}
else{
mycontinue*=0;
}
}

}//end of mycontinue

cout<<" ---now out, computing partial sums"<<endl;

myfileOUT.open(filenameOUT,ios::app);
myfileOUTTT.open(filenameOUTTT,ios::app);
for(u=0;u<M;u++){

if(u%50==0){
cout<<" computed "<<u<<" data points"<<endl;
}

//first output discrepancy and partial sums up to length u, and 11-th HAP
k=disc(datab,u);
if(u>0){
myfileOUTTT<<u<<"\t"<<k<<"\t";
for(d=1;d<u;d++){
if(d<=11){
myfileOUTTT<<glopartsum[d]<<"\t";
}
}
myfileOUTTT<<"\n";
}

//now output sequence value
if(datab[u]==1){
myfileOUT<<"+ ";
}
if(datab[u]==-1){
myfileOUT<<"- ";
}
}
myfileOUT.close();
myfileOUTTT.close();

cout<<" discrepancy="<<disc(datab,M)<<endl;

//exit normally
return 0;
}
</pre>

Refined greedy computation of multiplicative sequences

2010-01-14T21:01:15Z

Thomas:

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Back to the main experimental results page].

Here is the source of a C++ program for investigating refined greedy algorithms in the case of completely-multiplicative sequences.

The freedom lies in choosing the values at prime indices. Many methods can be devised but only a few are implemented for the moment, feel free to add many more (please indicate what you've added or modified). The code is commented and hopefully fairly straightforward (it probably could be optimized here and there, but has been carefully checked and seems correct).

The input file must be in the format of a sequence separated by spaces like + - + (with a space after the last sign, no carriage returns). These indicate the values at the first primes: <math>x_2</math>, <math>x_3</math>, <math>x_5</math>... In fact it is also possible not to specify any one value before the end by putting a 0.

There are three output files: the first one will contain the sequence found, the second may contain information on the choices made for each undetermined prime (I've commented it out for now), and the third contains in columns: n, D(n), partial sum of sequence up to n, partial sum of 2-HAP up to n, ... 11-HAP up to n (you can add more if so you wish).

Finally the constant M is the length of the sequence to be computed.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
13-jan-2010: written from scratch. Really basic code...
14-jan-2010: implemented various methods to choose values
at prime indices.
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------

int const M=3000;

int tab[M], primes[M], prtab[M], glopartsum[M];
char filenameIN[50], filenameOUT[50], filenameOUTT[50], filenameOUTTT[50];
ofstream myfileOUT;//sequence
ofstream myfileOUTT;//other data
ofstream myfileOUTTT;//yet other data
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//---------
//mypow
//integer power: a^b
//---------
int mypow(int a, int b){
int i, ans=1;
for(i=1;i<=b;i++){
ans*=a;
}
if(b==0){
return 1;
}
else{
return ans;
}
}

//---------
//primfact
//compute prime factors and store
//returns 2 iff n is prime
//---------
int primfact(int n){

int a, b, c, d, ncurr;
for(a=0;a<M;a++){
prtab[a]=0;
}
ncurr=n;
for(b=0;b<n;b++){//surely the n-th prime is greater than n already
while(primes[b]!=0 && ncurr>1 && (ncurr%primes[b])==0){
prtab[b]+=1;
ncurr/=primes[b];
}

if(primes[b]==0){
//exit
b+=M;
}
}
d=0;
for(c=0;c<M;c++){
if(prtab[c]!=0){
d+=prtab[c];
}
}
if(d==1){
//cout<<" "<<n<<" is prime"<<endl;
return 2;
}
if(d!=1){
//cout<<" "<<n<<" is composite"<<endl;
return 0;
}
}

//---------
//disc
//computes discrepancy of sequence of +/-1 contained in tabb
//over all HAPs up to len
//store signed partial sums up to len:
// glopartsum[0]=0, glopartsum[1]=sum of seq, glopartsum[2]=sum of 2-HAP...
//---------
int disc(int tabb[M], int len){

int n, d, i;
int signsum;
int absum;
int r;
int beflocdisc;
int aflocdisc;

beflocdisc=0;
aflocdisc=0;

for(n=2;n<=len;n++){
for(d=1;d<n;d++){
signsum=0;
absum=0;
glopartsum[d]=0;
r=int(floor(n/d));
for(i=1;i<=r;i++){
signsum+=tabb[d*i-1];
}
glopartsum[d]=signsum;
absum=abs(signsum);
aflocdisc=max(beflocdisc,absum);
beflocdisc=aflocdisc;
}
}
return aflocdisc;
}

//---------
//lplain
//just adds the entries of tabb
//---------
int lplain(int tabb[M], int len){
int norm, a;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[a];
}
return norm;
}

//---------
//l1norm
//---------
int l1norm(int tabb[M], int len){
int a, norm;
norm=0;
for(a=0;a<len;a++){
norm+=abs(tabb[a]);
}
return norm;
}

//---------
//l2norm
//---------
double l2norm(int tabb[M], int len){
int a, snorm;
double norm;
snorm=0; norm=0.0;
for(a=0;a<len;a++){
snorm+=(tabb[a]*tabb[a]);
}
norm=sqrt(snorm);
return norm;
}

//---------
//lharmw
//harmonic weighting
//---------
double lharmw(int tabb[M], int len){
int a;
double norm;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[M]/(a+1);
}
return norm;
}

//----------------------------------------------------------
//-- main:
//
//we compute a multiplicative sequence and its discrepancy by
//specifying its values at first p primes
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, w, j, c, t, imax=0, maxlength, currprime, iprec;
int datab[M];
int mycontinue, g, loclen, h;
double nmplus, nmminus;
string line, buff;
string myplus ("+");
string myminus ("-");
string myzero ("0");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output LONG MULT SEQUENCE filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

cout<<endl;
cout<<" give output CHOICE filename:"<<endl;
cin>>filenameOUTT;
cout<<endl;

cout<<endl;
cout<<" give output DISC & PARTSUM filename: "<<endl;
cin>>filenameOUTTT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

myfileOUTT.open(filenameOUTT,ios::out);
myfileOUTT.close();

myfileOUTTT.open(filenameOUTTT,ios::out);
myfileOUTTT.close();

//fill tables with zeros
for(i=0;i<M;i++){
tab[i];
datab[i]=0;
primes[i]=0;
prtab[i]=0;
}

//construct and store primes up to M
//primes[a] = (a+1)-th prime
iprec=2;
for(i=2;i<=M;i++){
k=0;
for(j=1;j<=i;j++){
if( (i%j)==0){
k++;
}
}
if(k==2){
primes[iprec-2]=i;
iprec+=1;
}
}

//-- read the choices of values at first few primes
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;
getline(myfileIN,line);
stringstream stsm(line);
while(stsm>>buff){
currprime=primes[i];
if(buff.compare(myplus)==0){
datab[currprime-1]=1;
}
if(buff.compare(myminus)==0){
datab[currprime-1]=-1;
}
if(buff.compare(myzero)==0){
datab[currprime-1]=0;
}
if( datab[currprime-1] !=1 && datab[currprime-1] !=-1 && datab[currprime-1]!=0){
cout<<"problem: element number "<<i+1<<" is neither + nor - nor 0."<<endl;
abort();
}
i+=1;
}
imax=i;
cout<<" this initial sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}
myfileIN.close();

//-- main processing:
//fill the rest of the sequence multiplicatively
//if choices are to be made, take +1 but tell the user

//convention: datab[i]=x_{i+1}
//and first index is always associated to +
datab[0]=1;

mycontinue=0;

while(mycontinue==0){

for(u=2;u<M;u++){

if(datab[u-1]==0){

k=0;
k=primfact(u);

//if prime: there's freedom, choose one method to fill it
if(k==2){

//choice 1: silly uniform -1
//datab[u-1]=-1;

//choice 2: take into account of partial sums
loclen=u;

cout<<" adressing "<<loclen<<endl;

//values are known below loclen, not at prime loclen,
//and from loclen+1 to loclen+h. we then choose loclen
//so as to minimize most partial HAP sums (lots of flexibility)

//compute h
h=1;
for(v=loclen+1;v<M;v++){
if(datab[v]==0){
h=v-loclen;
v+=M;
}
}
cout<<" values after are known up to "<<loclen+h<<endl;

//now compute the effect of imposing datab[u-1]=+1
//would have on partial sums up to loclen+h-1
datab[u-1]=+1;
c=disc(datab,loclen+h-1);
nmplus=lplain(datab,loclen+h-1);

//compute the same with -1 instead
datab[u-1]=-1;
c=disc(datab,loclen+h-1);
nmminus=lplain(datab,loclen+h-1);

//now choose
if(abs(nmplus)>abs(nmminus)){
datab[u-1]=-1;
}
else{
datab[u-1]=+1;
}
cout<<" nmplus="<<nmplus<<", nmminus="<<nmminus<<endl;

/*------temporarily removed, could be useful to monitor
//record those numbers for further analysis
myfileOUTT.open(filenameOUTT,ios::app);
myfileOUTT<<u<<"\t"<<nmplus<<"\t"<<nmminus<<endl;
myfileOUTT.close();
------------------------*/

//choice 3:
//more ideas to be added...

//in any case: give info
cout<<" imposing: x_"<<u<<"="<<datab[u-1]<<endl;

}
else{//if composite then try to compute it multiplicatively

//loop on prime factors of u to see what is known so far

w=1;
for(j=0;j<M;j++){

if(prtab[j]!=0 && datab[ primes[j]-1 ]!=0){
//then this x_j is known
w*=1;
}

if(prtab[j]!=0 && datab[ primes[j]-1 ]==0){
//then this x_j not known, enough to discard
w*=0;
}
}

//if all x_j are known then compute x_u
if(w==1){
datab[u-1]=1;
for(j=0;j<M;j++){
if(prtab[j]!=0){
datab[u-1]*=mypow(datab[ primes[j]-1 ],prtab[j]);
}
}
cout<<" just computed x_"<<u<<"="<<datab[u-1]<<endl;
}
else{
//cout<<" cannot yet compute x_"<<u<<endl;
//cout<<endl;
}

}//end of mult comp attempt
}//end of this u
else{
//cout<<" already known"<<endl;
}
}//end of u loop

//now see whether we should go on
mycontinue=1;
for(c=0;c<M-1;c++){
if(datab[c]!=0){
mycontinue*=1;
}
else{
mycontinue*=0;
}
}

}//end of mycontinue

cout<<" ---now out, computing partial sums"<<endl;

myfileOUT.open(filenameOUT,ios::app);
myfileOUTTT.open(filenameOUTTT,ios::app);
for(u=0;u<M;u++){

if(u%50==0){
cout<<" computed "<<u<<" data points"<<endl;
}

//first output discrepancy and partial sums up to length u, and 11-th HAP
k=disc(datab,u);
if(u>0){
myfileOUTTT<<u<<"\t"<<k<<"\t";
for(d=1;d<u;d++){
if(d<=11){
myfileOUTTT<<glopartsum[d]<<"\t";
}
}
myfileOUTTT<<"\n";
}

//now output sequence value
if(datab[u]==1){
myfileOUT<<"+ ";
}
if(datab[u]==-1){
myfileOUT<<"- ";
}
}
myfileOUT.close();
myfileOUTTT.close();

cout<<" discrepancy="<<disc(datab,M)<<endl;

//exit normally
return 0;
}
</pre>

Refined greedy computation of multiplicative sequences

2010-01-14T21:00:18Z

Thomas:

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Back to the main experimental results page].

Here is the source of a C++ program for investigating refined greedy algorithms in the case of completely-multiplicative sequences.

The freedom lies in choosing the values at prime indices. Many methods can be devised but only a few are implemented for the moment, feel free to add many more (please indicate what you've added or modified). The code is commented and hopefully fairly straightforward (it probably could be optimized here and there, but has been carefully checked and seems correct).

The input file must be in the format of a sequence separated by spaces like + - + (with a space after the last sign, no carriage returns). These indicate the values at the first primes: <math>x_2</math>, <math>x_3</math>, <math>x_5</math>... In fact it is also possible not to specify any one value before the end by putting a 0.

There are three output files: the first one will contain the sequence found, the second may contain information on the choices made for each undetermined prime (I've commented it out for now), and the third contains in columns: n, D(n), partial sum of sequence up to n, partial sum of 2-HAP up to n, ... 11-HAP up to n (you can add more if so you wich).

Finally the constant M is the length of the sequence to be computed.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
13-jan-2010: written from scratch. Really basic code...
14-jan-2010: implemented various methods to choose values
at prime indices.
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------

int const M=3000;

int tab[M], primes[M], prtab[M], glopartsum[M];
char filenameIN[50], filenameOUT[50], filenameOUTT[50], filenameOUTTT[50];
ofstream myfileOUT;//sequence
ofstream myfileOUTT;//other data
ofstream myfileOUTTT;//yet other data
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//---------
//mypow
//integer power: a^b
//---------
int mypow(int a, int b){
int i, ans=1;
for(i=1;i<=b;i++){
ans*=a;
}
if(b==0){
return 1;
}
else{
return ans;
}
}

//---------
//primfact
//compute prime factors and store
//returns 2 iff n is prime
//---------
int primfact(int n){

int a, b, c, d, ncurr;
for(a=0;a<M;a++){
prtab[a]=0;
}
ncurr=n;
for(b=0;b<n;b++){//surely the n-th prime is greater than n already
while(primes[b]!=0 && ncurr>1 && (ncurr%primes[b])==0){
prtab[b]+=1;
ncurr/=primes[b];
}

if(primes[b]==0){
//exit
b+=M;
}
}
d=0;
for(c=0;c<M;c++){
if(prtab[c]!=0){
d+=prtab[c];
}
}
if(d==1){
//cout<<" "<<n<<" is prime"<<endl;
return 2;
}
if(d!=1){
//cout<<" "<<n<<" is composite"<<endl;
return 0;
}
}

//---------
//disc
//computes discrepancy of sequence of +/-1 contained in tabb
//over all HAPs up to len
//store signed partial sums up to len:
// glopartsum[0]=0, glopartsum[1]=sum of seq, glopartsum[2]=sum of 2-HAP...
//---------
int disc(int tabb[M], int len){

int n, d, i;
int signsum;
int absum;
int r;
int beflocdisc;
int aflocdisc;

beflocdisc=0;
aflocdisc=0;

for(n=2;n<=len;n++){
for(d=1;d<n;d++){
signsum=0;
absum=0;
glopartsum[d]=0;
r=int(floor(n/d));
for(i=1;i<=r;i++){
signsum+=tabb[d*i-1];
}
glopartsum[d]=signsum;
absum=abs(signsum);
aflocdisc=max(beflocdisc,absum);
beflocdisc=aflocdisc;
}
}
return aflocdisc;
}

//---------
//lplain
//just adds the entries of tabb
//---------
int lplain(int tabb[M], int len){
int norm, a;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[a];
}
return norm;
}

//---------
//l1norm
//---------
int l1norm(int tabb[M], int len){
int a, norm;
norm=0;
for(a=0;a<len;a++){
norm+=abs(tabb[a]);
}
return norm;
}

//---------
//l2norm
//---------
double l2norm(int tabb[M], int len){
int a, snorm;
double norm;
snorm=0; norm=0.0;
for(a=0;a<len;a++){
snorm+=(tabb[a]*tabb[a]);
}
norm=sqrt(snorm);
return norm;
}

//---------
//lharmw
//harmonic weighting
//---------
double lharmw(int tabb[M], int len){
int a;
double norm;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[M]/(a+1);
}
return norm;
}

//----------------------------------------------------------
//-- main:
//
//we compute a multiplicative sequence and its discrepancy by
//specifying its values at first p primes
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, w, j, c, t, imax=0, maxlength, currprime, iprec;
int datab[M];
int mycontinue, g, loclen, h;
double nmplus, nmminus;
string line, buff;
string myplus ("+");
string myminus ("-");
string myzero ("0");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output LONG MULT SEQUENCE filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

cout<<endl;
cout<<" give output CHOICE filename:"<<endl;
cin>>filenameOUTT;
cout<<endl;

cout<<endl;
cout<<" give output DISC & PARTSUM filename: "<<endl;
cin>>filenameOUTTT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

myfileOUTT.open(filenameOUTT,ios::out);
myfileOUTT.close();

myfileOUTTT.open(filenameOUTTT,ios::out);
myfileOUTTT.close();

//fill tables with zeros
for(i=0;i<M;i++){
tab[i];
datab[i]=0;
primes[i]=0;
prtab[i]=0;
}

//construct and store primes up to M
//primes[a] = (a+1)-th prime
iprec=2;
for(i=2;i<=M;i++){
k=0;
for(j=1;j<=i;j++){
if( (i%j)==0){
k++;
}
}
if(k==2){
primes[iprec-2]=i;
iprec+=1;
}
}

//-- read the choices of values at first few primes
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;
getline(myfileIN,line);
stringstream stsm(line);
while(stsm>>buff){
currprime=primes[i];
if(buff.compare(myplus)==0){
datab[currprime-1]=1;
}
if(buff.compare(myminus)==0){
datab[currprime-1]=-1;
}
if(buff.compare(myzero)==0){
datab[currprime-1]=0;
}
if( datab[currprime-1] !=1 && datab[currprime-1] !=-1 && datab[currprime-1]!=0){
cout<<"problem: element number "<<i+1<<" is neither + nor - nor 0."<<endl;
abort();
}
i+=1;
}
imax=i;
cout<<" this initial sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}
myfileIN.close();

//-- main processing:
//fill the rest of the sequence multiplicatively
//if choices are to be made, take +1 but tell the user

//convention: datab[i]=x_{i+1}
//and first index is always associated to +
datab[0]=1;

mycontinue=0;

while(mycontinue==0){

for(u=2;u<M;u++){

if(datab[u-1]==0){

k=0;
k=primfact(u);

//if prime: there's freedom, choose one method to fill it
if(k==2){

//choice 1: silly uniform -1
//datab[u-1]=-1;

//choice 2: take into account of partial sums
loclen=u;

cout<<" adressing "<<loclen<<endl;

//values are known below loclen, not at prime loclen,
//and from loclen+1 to loclen+h. we then choose loclen
//so as to minimize most partial HAP sums (lots of flexibility)

//compute h
h=1;
for(v=loclen+1;v<M;v++){
if(datab[v]==0){
h=v-loclen;
v+=M;
}
}
cout<<" values after are known up to "<<loclen+h<<endl;

//now compute the effect of imposing datab[u-1]=+1
//would have on partial sums up to loclen+h-1
datab[u-1]=+1;
c=disc(datab,loclen+h-1);
nmplus=lplain(datab,loclen+h-1);

//compute the same with -1 instead
datab[u-1]=-1;
c=disc(datab,loclen+h-1);
nmminus=lplain(datab,loclen+h-1);

//now choose
if(abs(nmplus)>abs(nmminus)){
datab[u-1]=-1;
}
else{
datab[u-1]=+1;
}
cout<<" nmplus="<<nmplus<<", nmminus="<<nmminus<<endl;

/*------temporarily removed, could be useful to monitor
//record those numbers for further analysis
myfileOUTT.open(filenameOUTT,ios::app);
myfileOUTT<<u<<"\t"<<nmplus<<"\t"<<nmminus<<endl;
myfileOUTT.close();
------------------------*/

//choice 3:
//more ideas to be added...

//in any case: give info
cout<<" imposing: x_"<<u<<"="<<datab[u-1]<<endl;

}
else{//if composite then try to compute it multiplicatively

//loop on prime factors of u to see what is known so far

w=1;
for(j=0;j<M;j++){

if(prtab[j]!=0 && datab[ primes[j]-1 ]!=0){
//then this x_j is known
w*=1;
}

if(prtab[j]!=0 && datab[ primes[j]-1 ]==0){
//then this x_j not known, enough to discard
w*=0;
}
}

//if all x_j are known then compute x_u
if(w==1){
datab[u-1]=1;
for(j=0;j<M;j++){
if(prtab[j]!=0){
datab[u-1]*=mypow(datab[ primes[j]-1 ],prtab[j]);
}
}
cout<<" just computed x_"<<u<<"="<<datab[u-1]<<endl;
}
else{
//cout<<" cannot yet compute x_"<<u<<endl;
//cout<<endl;
}

}//end of mult comp attempt
}//end of this u
else{
//cout<<" already known"<<endl;
}
}//end of u loop

//now see whether we should go on
mycontinue=1;
for(c=0;c<M-1;c++){
if(datab[c]!=0){
mycontinue*=1;
}
else{
mycontinue*=0;
}
}

}//end of mycontinue

cout<<" ---now out, computing partial sums"<<endl;

myfileOUT.open(filenameOUT,ios::app);
myfileOUTTT.open(filenameOUTTT,ios::app);
for(u=0;u<M;u++){

if(u%50==0){
cout<<" computed "<<u<<" data points"<<endl;
}

//first output discrepancy and partial sums up to length u, and 11-th HAP
k=disc(datab,u);
if(u>0){
myfileOUTTT<<u<<"\t"<<k<<"\t";
for(d=1;d<u;d++){
if(d<=11){
myfileOUTTT<<glopartsum[d]<<"\t";
}
}
myfileOUTTT<<"\n";
}

//now output sequence value
if(datab[u]==1){
myfileOUT<<"+ ";
}
if(datab[u]==-1){
myfileOUT<<"- ";
}
}
myfileOUT.close();
myfileOUTTT.close();

cout<<" discrepancy="<<disc(datab,M)<<endl;

//exit normally
return 0;
}
</pre>

Multiplicative sequences

2010-01-14T20:58:17Z

Thomas:

Multiplicative sequences

2010-01-14T20:47:10Z

Thomas: added data on sum of partial sums method

Refined greedy computation of multiplicative sequences

2010-01-14T20:37:23Z

Thomas: added link to exp page

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Back to the main experimental results page].

Here is the source of a C++ program for investigating refined greedy algorithms in the case of completely-multiplicative sequences.

The freedom lies in choosing the values at prime indices. Many methods can be devised but only a few are implemented for the moment, feel free to add many more (please indicate what you've added or modified). The code is commented and hopefully fairly straightforward (it probably could be optimized here and there, but has been carefully checked and seems correct).

The input file must be in the format of a sequence separated by spaces like + - + (with a space after the last sign, no carriage returns). These indicate the values at the first primes: <math>x_2</math>, <math>x_3</math>, <math>x_5</math>... In fact it is also possible not to specify any value before the end by putting a 0.

There are three output files: the first one will contain the sequence found, the second may contain information on the choices made for each undetermined prime (I've commented it out for now), and the third contains in columns: n, D(n), partial sum of sequence up to n, partial sum of 2-HAP up to n, ... 11-HAP up to n (you can add more if so you wich).

Finally the constant M is the length of the sequence to be computed.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
13-jan-2010: written from scratch. Really basic code...
14-jan-2010: implemented various methods to choose values
at prime indices.
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------

int const M=3000;

int tab[M], primes[M], prtab[M], glopartsum[M];
char filenameIN[50], filenameOUT[50], filenameOUTT[50], filenameOUTTT[50];
ofstream myfileOUT;//sequence
ofstream myfileOUTT;//other data
ofstream myfileOUTTT;//yet other data
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//---------
//mypow
//integer power: a^b
//---------
int mypow(int a, int b){
int i, ans=1;
for(i=1;i<=b;i++){
ans*=a;
}
if(b==0){
return 1;
}
else{
return ans;
}
}

//---------
//primfact
//compute prime factors and store
//returns 2 iff n is prime
//---------
int primfact(int n){

int a, b, c, d, ncurr;
for(a=0;a<M;a++){
prtab[a]=0;
}
ncurr=n;
for(b=0;b<n;b++){//surely the n-th prime is greater than n already
while(primes[b]!=0 && ncurr>1 && (ncurr%primes[b])==0){
prtab[b]+=1;
ncurr/=primes[b];
}

if(primes[b]==0){
//exit
b+=M;
}
}
d=0;
for(c=0;c<M;c++){
if(prtab[c]!=0){
d+=prtab[c];
}
}
if(d==1){
//cout<<" "<<n<<" is prime"<<endl;
return 2;
}
if(d!=1){
//cout<<" "<<n<<" is composite"<<endl;
return 0;
}
}

//---------
//disc
//computes discrepancy of sequence of +/-1 contained in tabb
//over all HAPs up to len
//store signed partial sums up to len:
// glopartsum[0]=0, glopartsum[1]=sum of seq, glopartsum[2]=sum of 2-HAP...
//---------
int disc(int tabb[M], int len){

int n, d, i;
int signsum;
int absum;
int r;
int beflocdisc;
int aflocdisc;

beflocdisc=0;
aflocdisc=0;

for(n=2;n<=len;n++){
for(d=1;d<n;d++){
signsum=0;
absum=0;
glopartsum[d]=0;
r=int(floor(n/d));
for(i=1;i<=r;i++){
signsum+=tabb[d*i-1];
}
glopartsum[d]=signsum;
absum=abs(signsum);
aflocdisc=max(beflocdisc,absum);
beflocdisc=aflocdisc;
}
}
return aflocdisc;
}

//---------
//lplain
//just adds the entries of tabb
//---------
int lplain(int tabb[M], int len){
int norm, a;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[a];
}
return norm;
}

//---------
//l1norm
//---------
int l1norm(int tabb[M], int len){
int a, norm;
norm=0;
for(a=0;a<len;a++){
norm+=abs(tabb[a]);
}
return norm;
}

//---------
//l2norm
//---------
double l2norm(int tabb[M], int len){
int a, snorm;
double norm;
snorm=0; norm=0.0;
for(a=0;a<len;a++){
snorm+=(tabb[a]*tabb[a]);
}
norm=sqrt(snorm);
return norm;
}

//---------
//lharmw
//harmonic weighting
//---------
double lharmw(int tabb[M], int len){
int a;
double norm;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[M]/(a+1);
}
return norm;
}

//----------------------------------------------------------
//-- main:
//
//we compute a multiplicative sequence and its discrepancy by
//specifying its values at first p primes
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, w, j, c, t, imax=0, maxlength, currprime, iprec;
int datab[M];
int mycontinue, g, loclen, h;
double nmplus, nmminus;
string line, buff;
string myplus ("+");
string myminus ("-");
string myzero ("0");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output LONG MULT SEQUENCE filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

cout<<endl;
cout<<" give output CHOICE filename:"<<endl;
cin>>filenameOUTT;
cout<<endl;

cout<<endl;
cout<<" give output DISC & PARTSUM filename: "<<endl;
cin>>filenameOUTTT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

myfileOUTT.open(filenameOUTT,ios::out);
myfileOUTT.close();

myfileOUTTT.open(filenameOUTTT,ios::out);
myfileOUTTT.close();

//fill tables with zeros
for(i=0;i<M;i++){
tab[i];
datab[i]=0;
primes[i]=0;
prtab[i]=0;
}

//construct and store primes up to M
//primes[a] = (a+1)-th prime
iprec=2;
for(i=2;i<=M;i++){
k=0;
for(j=1;j<=i;j++){
if( (i%j)==0){
k++;
}
}
if(k==2){
primes[iprec-2]=i;
iprec+=1;
}
}

//-- read the choices of values at first few primes
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;
getline(myfileIN,line);
stringstream stsm(line);
while(stsm>>buff){
currprime=primes[i];
if(buff.compare(myplus)==0){
datab[currprime-1]=1;
}
if(buff.compare(myminus)==0){
datab[currprime-1]=-1;
}
if(buff.compare(myzero)==0){
datab[currprime-1]=0;
}
if( datab[currprime-1] !=1 && datab[currprime-1] !=-1 && datab[currprime-1]!=0){
cout<<"problem: element number "<<i+1<<" is neither + nor - nor 0."<<endl;
abort();
}
i+=1;
}
imax=i;
cout<<" this initial sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}
myfileIN.close();

//-- main processing:
//fill the rest of the sequence multiplicatively
//if choices are to be made, take +1 but tell the user

//convention: datab[i]=x_{i+1}
//and first index is always associated to +
datab[0]=1;

mycontinue=0;

while(mycontinue==0){

for(u=2;u<M;u++){

if(datab[u-1]==0){

k=0;
k=primfact(u);

//if prime: there's freedom, choose one method to fill it
if(k==2){

//choice 1: silly uniform -1
//datab[u-1]=-1;

//choice 2: take into account of partial sums
loclen=u;

cout<<" adressing "<<loclen<<endl;

//values are known below loclen, not at prime loclen,
//and from loclen+1 to loclen+h. we then choose loclen
//so as to minimize most partial HAP sums (lots of flexibility)

//compute h
h=1;
for(v=loclen+1;v<M;v++){
if(datab[v]==0){
h=v-loclen;
v+=M;
}
}
cout<<" values after are known up to "<<loclen+h<<endl;

//now compute the effect of imposing datab[u-1]=+1
//would have on partial sums up to loclen+h-1
datab[u-1]=+1;
c=disc(datab,loclen+h-1);
nmplus=lplain(datab,loclen+h-1);

//compute the same with -1 instead
datab[u-1]=-1;
c=disc(datab,loclen+h-1);
nmminus=lplain(datab,loclen+h-1);

//now choose
if(abs(nmplus)>abs(nmminus)){
datab[u-1]=-1;
}
else{
datab[u-1]=+1;
}
cout<<" nmplus="<<nmplus<<", nmminus="<<nmminus<<endl;

/*------temporarily removed, could be useful to monitor
//record those numbers for further analysis
myfileOUTT.open(filenameOUTT,ios::app);
myfileOUTT<<u<<"\t"<<nmplus<<"\t"<<nmminus<<endl;
myfileOUTT.close();
------------------------*/

//choice 3:
//more ideas to be added...

//in any case: give info
cout<<" imposing: x_"<<u<<"="<<datab[u-1]<<endl;

}
else{//if composite then try to compute it multiplicatively

//loop on prime factors of u to see what is known so far

w=1;
for(j=0;j<M;j++){

if(prtab[j]!=0 && datab[ primes[j]-1 ]!=0){
//then this x_j is known
w*=1;
}

if(prtab[j]!=0 && datab[ primes[j]-1 ]==0){
//then this x_j not known, enough to discard
w*=0;
}
}

//if all x_j are known then compute x_u
if(w==1){
datab[u-1]=1;
for(j=0;j<M;j++){
if(prtab[j]!=0){
datab[u-1]*=mypow(datab[ primes[j]-1 ],prtab[j]);
}
}
cout<<" just computed x_"<<u<<"="<<datab[u-1]<<endl;
}
else{
//cout<<" cannot yet compute x_"<<u<<endl;
//cout<<endl;
}

}//end of mult comp attempt
}//end of this u
else{
//cout<<" already known"<<endl;
}
}//end of u loop

//now see whether we should go on
mycontinue=1;
for(c=0;c<M-1;c++){
if(datab[c]!=0){
mycontinue*=1;
}
else{
mycontinue*=0;
}
}

}//end of mycontinue

cout<<" ---now out, computing partial sums"<<endl;

myfileOUT.open(filenameOUT,ios::app);
myfileOUTTT.open(filenameOUTTT,ios::app);
for(u=0;u<M;u++){

if(u%50==0){
cout<<" computed "<<u<<" data points"<<endl;
}

//first output discrepancy and partial sums up to length u, and 11-th HAP
k=disc(datab,u);
if(u>0){
myfileOUTTT<<u<<"\t"<<k<<"\t";
for(d=1;d<u;d++){
if(d<=11){
myfileOUTTT<<glopartsum[d]<<"\t";
}
}
myfileOUTTT<<"\n";
}

//now output sequence value
if(datab[u]==1){
myfileOUT<<"+ ";
}
if(datab[u]==-1){
myfileOUT<<"- ";
}
}
myfileOUT.close();
myfileOUTTT.close();

cout<<" discrepancy="<<disc(datab,M)<<endl;

//exit normally
return 0;
}
</pre>

Refined greedy computation of multiplicative sequences

2010-01-14T20:34:45Z

Thomas: put up the code and described it

Here is the source of a C++ program for investigating refined greedy algorithms in the case of completely-multiplicative sequences.

The freedom lies in choosing the values at prime indices. Many methods can be devised but only a few are implemented for the moment, feel free to add many more (please indicate what you've added or modified). The code is commented and hopefully fairly straightforward (it probably could be optimized here and there, but has been carefully checked and seems correct).

The input file must be in the format of a sequence separated by spaces like + - + (with a space after the last sign, no carriage returns). These indicate the values at the first primes: <math>x_2</math>, <math>x_3</math>, <math>x_5</math>... In fact it is also possible not to specify any value before the end by putting a 0.

There are three output files: the first one will contain the sequence found, the second may contain information on the choices made for each undetermined prime (I've commented it out for now), and the third contains in columns: n, D(n), partial sum of sequence up to n, partial sum of 2-HAP up to n, ... 11-HAP up to n (you can add more if so you wich).

Finally the constant M is the length of the sequence to be computed.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
13-jan-2010: written from scratch. Really basic code...
14-jan-2010: implemented various methods to choose values
at prime indices.
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------

int const M=3000;

int tab[M], primes[M], prtab[M], glopartsum[M];
char filenameIN[50], filenameOUT[50], filenameOUTT[50], filenameOUTTT[50];
ofstream myfileOUT;//sequence
ofstream myfileOUTT;//other data
ofstream myfileOUTTT;//yet other data
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//---------
//mypow
//integer power: a^b
//---------
int mypow(int a, int b){
int i, ans=1;
for(i=1;i<=b;i++){
ans*=a;
}
if(b==0){
return 1;
}
else{
return ans;
}
}

//---------
//primfact
//compute prime factors and store
//returns 2 iff n is prime
//---------
int primfact(int n){

int a, b, c, d, ncurr;
for(a=0;a<M;a++){
prtab[a]=0;
}
ncurr=n;
for(b=0;b<n;b++){//surely the n-th prime is greater than n already
while(primes[b]!=0 && ncurr>1 && (ncurr%primes[b])==0){
prtab[b]+=1;
ncurr/=primes[b];
}

if(primes[b]==0){
//exit
b+=M;
}
}
d=0;
for(c=0;c<M;c++){
if(prtab[c]!=0){
d+=prtab[c];
}
}
if(d==1){
//cout<<" "<<n<<" is prime"<<endl;
return 2;
}
if(d!=1){
//cout<<" "<<n<<" is composite"<<endl;
return 0;
}
}

//---------
//disc
//computes discrepancy of sequence of +/-1 contained in tabb
//over all HAPs up to len
//store signed partial sums up to len:
// glopartsum[0]=0, glopartsum[1]=sum of seq, glopartsum[2]=sum of 2-HAP...
//---------
int disc(int tabb[M], int len){

int n, d, i;
int signsum;
int absum;
int r;
int beflocdisc;
int aflocdisc;

beflocdisc=0;
aflocdisc=0;

for(n=2;n<=len;n++){
for(d=1;d<n;d++){
signsum=0;
absum=0;
glopartsum[d]=0;
r=int(floor(n/d));
for(i=1;i<=r;i++){
signsum+=tabb[d*i-1];
}
glopartsum[d]=signsum;
absum=abs(signsum);
aflocdisc=max(beflocdisc,absum);
beflocdisc=aflocdisc;
}
}
return aflocdisc;
}

//---------
//lplain
//just adds the entries of tabb
//---------
int lplain(int tabb[M], int len){
int norm, a;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[a];
}
return norm;
}

//---------
//l1norm
//---------
int l1norm(int tabb[M], int len){
int a, norm;
norm=0;
for(a=0;a<len;a++){
norm+=abs(tabb[a]);
}
return norm;
}

//---------
//l2norm
//---------
double l2norm(int tabb[M], int len){
int a, snorm;
double norm;
snorm=0; norm=0.0;
for(a=0;a<len;a++){
snorm+=(tabb[a]*tabb[a]);
}
norm=sqrt(snorm);
return norm;
}

//---------
//lharmw
//harmonic weighting
//---------
double lharmw(int tabb[M], int len){
int a;
double norm;
norm=0;
for(a=0;a<len;a++){
norm+=tabb[M]/(a+1);
}
return norm;
}

//----------------------------------------------------------
//-- main:
//
//we compute a multiplicative sequence and its discrepancy by
//specifying its values at first p primes
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, w, j, c, t, imax=0, maxlength, currprime, iprec;
int datab[M];
int mycontinue, g, loclen, h;
double nmplus, nmminus;
string line, buff;
string myplus ("+");
string myminus ("-");
string myzero ("0");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output LONG MULT SEQUENCE filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

cout<<endl;
cout<<" give output CHOICE filename:"<<endl;
cin>>filenameOUTT;
cout<<endl;

cout<<endl;
cout<<" give output DISC & PARTSUM filename: "<<endl;
cin>>filenameOUTTT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

myfileOUTT.open(filenameOUTT,ios::out);
myfileOUTT.close();

myfileOUTTT.open(filenameOUTTT,ios::out);
myfileOUTTT.close();

//fill tables with zeros
for(i=0;i<M;i++){
tab[i];
datab[i]=0;
primes[i]=0;
prtab[i]=0;
}

//construct and store primes up to M
//primes[a] = (a+1)-th prime
iprec=2;
for(i=2;i<=M;i++){
k=0;
for(j=1;j<=i;j++){
if( (i%j)==0){
k++;
}
}
if(k==2){
primes[iprec-2]=i;
iprec+=1;
}
}

//-- read the choices of values at first few primes
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;
getline(myfileIN,line);
stringstream stsm(line);
while(stsm>>buff){
currprime=primes[i];
if(buff.compare(myplus)==0){
datab[currprime-1]=1;
}
if(buff.compare(myminus)==0){
datab[currprime-1]=-1;
}
if(buff.compare(myzero)==0){
datab[currprime-1]=0;
}
if( datab[currprime-1] !=1 && datab[currprime-1] !=-1 && datab[currprime-1]!=0){
cout<<"problem: element number "<<i+1<<" is neither + nor - nor 0."<<endl;
abort();
}
i+=1;
}
imax=i;
cout<<" this initial sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}
myfileIN.close();

//-- main processing:
//fill the rest of the sequence multiplicatively
//if choices are to be made, take +1 but tell the user

//convention: datab[i]=x_{i+1}
//and first index is always associated to +
datab[0]=1;

mycontinue=0;

while(mycontinue==0){

for(u=2;u<M;u++){

if(datab[u-1]==0){

k=0;
k=primfact(u);

//if prime: there's freedom, choose one method to fill it
if(k==2){

//choice 1: silly uniform -1
//datab[u-1]=-1;

//choice 2: take into account of partial sums
loclen=u;

cout<<" adressing "<<loclen<<endl;

//values are known below loclen, not at prime loclen,
//and from loclen+1 to loclen+h. we then choose loclen
//so as to minimize most partial HAP sums (lots of flexibility)

//compute h
h=1;
for(v=loclen+1;v<M;v++){
if(datab[v]==0){
h=v-loclen;
v+=M;
}
}
cout<<" values after are known up to "<<loclen+h<<endl;

//now compute the effect of imposing datab[u-1]=+1
//would have on partial sums up to loclen+h-1
datab[u-1]=+1;
c=disc(datab,loclen+h-1);
nmplus=lplain(datab,loclen+h-1);

//compute the same with -1 instead
datab[u-1]=-1;
c=disc(datab,loclen+h-1);
nmminus=lplain(datab,loclen+h-1);

//now choose
if(abs(nmplus)>abs(nmminus)){
datab[u-1]=-1;
}
else{
datab[u-1]=+1;
}
cout<<" nmplus="<<nmplus<<", nmminus="<<nmminus<<endl;

/*------temporarily removed, could be useful to monitor
//record those numbers for further analysis
myfileOUTT.open(filenameOUTT,ios::app);
myfileOUTT<<u<<"\t"<<nmplus<<"\t"<<nmminus<<endl;
myfileOUTT.close();
------------------------*/

//choice 3:
//more ideas to be added...

//in any case: give info
cout<<" imposing: x_"<<u<<"="<<datab[u-1]<<endl;

}
else{//if composite then try to compute it multiplicatively

//loop on prime factors of u to see what is known so far

w=1;
for(j=0;j<M;j++){

if(prtab[j]!=0 && datab[ primes[j]-1 ]!=0){
//then this x_j is known
w*=1;
}

if(prtab[j]!=0 && datab[ primes[j]-1 ]==0){
//then this x_j not known, enough to discard
w*=0;
}
}

//if all x_j are known then compute x_u
if(w==1){
datab[u-1]=1;
for(j=0;j<M;j++){
if(prtab[j]!=0){
datab[u-1]*=mypow(datab[ primes[j]-1 ],prtab[j]);
}
}
cout<<" just computed x_"<<u<<"="<<datab[u-1]<<endl;
}
else{
//cout<<" cannot yet compute x_"<<u<<endl;
//cout<<endl;
}

}//end of mult comp attempt
}//end of this u
else{
//cout<<" already known"<<endl;
}
}//end of u loop

//now see whether we should go on
mycontinue=1;
for(c=0;c<M-1;c++){
if(datab[c]!=0){
mycontinue*=1;
}
else{
mycontinue*=0;
}
}

}//end of mycontinue

cout<<" ---now out, computing partial sums"<<endl;

myfileOUT.open(filenameOUT,ios::app);
myfileOUTTT.open(filenameOUTTT,ios::app);
for(u=0;u<M;u++){

if(u%50==0){
cout<<" computed "<<u<<" data points"<<endl;
}

//first output discrepancy and partial sums up to length u, and 11-th HAP
k=disc(datab,u);
if(u>0){
myfileOUTTT<<u<<"\t"<<k<<"\t";
for(d=1;d<u;d++){
if(d<=11){
myfileOUTTT<<glopartsum[d]<<"\t";
}
}
myfileOUTTT<<"\n";
}

//now output sequence value
if(datab[u]==1){
myfileOUT<<"+ ";
}
if(datab[u]==-1){
myfileOUT<<"- ";
}
}
myfileOUT.close();
myfileOUTTT.close();

cout<<" discrepancy="<<disc(datab,M)<<endl;

//exit normally
return 0;
}
</pre>

Experimental results

2010-01-14T20:17:35Z

Thomas: added link to source on refined greedy methods

[[The Erdős discrepancy problem|To return to the main Polymath5 page, click here]].

Perhaps we should have two kinds of subpages to this page: Pages about finding examples, and pages about analyzing them?

== Experimental data==
* [[The first 1124-sequence]] with discrepancy 2. ''Some more description''
* Other [[length 1124 sequences]] with discrepancy 2. ''Some more description''
* A [[sequence of length 1112]] derived from one with nice multiplicative properties.
* Some data about the problem with [[different upper and lower bound]]. Let N(a,b) be the largest N such that there is a sequence <math>x_1,\dots,x_N</math> all of whose HAP-errors are between -a and b, inclusive.
* Sequences taking values in <math>\mathbb{T}</math>:
** [[4th roots of unity]]
** [[6th roots of unity]]
* [http://thomas1111.wordpress.com/2010/01/10/tables-for-a-c10-candidate/ A sequence of length 407] with discrepancy 2 such that <math>x_n=x_{32 n}</math> for every n. [[The HAP-subsequence structure of that sequence]].
* More [[T32-invariant sequences]].
* Long [[multiplicative sequences]].
* Long sequences satisfying [[T2(x) = -x]].
* Long sequences satisfying [[T2(x) = T5(x) = -x]]
* Long sequences satisfying [[T2(x) = -T3(x)]].
* Long sequences satisfying constraints of the form [[T_m(x) = (+/-)T_n(x)]].
* Table of [[longest constrained sequences]].
* Table of [[short sequences statistics]].

==Source code==

* [[Convert raw input string into CSV table]]
* [[Create tables in an HTML file from an input sequence]]
* [[Verify the bounded discrepancy of an input sequence]]
* [[Depth-first search]]
* [[Search for completely multiplicative sequences]]
* [[Refined greedy computation of multiplicative sequences]]

==Wish list==

There is a separate page for [[proposals for finding long low-discrepancy sequences]]. It goes without saying that implementing any of these proposals belongs to the wish list.


* Find long/longest quasi-multiplicative sequences with some fixed group G, function <math>G\to \{-1,1\}</math> and maximal discrepancy C
** <math>G=C_6</math> and the function that sends 0,1 and 2 to 1 (because this seems to be a good choice)
* Do a "Mark-Bennet-style analysis" of one of the new 1124-sequences. [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4827] Also [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4842 done] (by Mark Bennet).
*. Take a moderately large k and search for the longest sequence of discrepancy 2 that's constructed as follows. First, pick a completely multiplicative function f to the group <math>C_{2k}</math>. Then set <math>x_n</math> to be 1 if f(n) lies between 0 and k-1, and -1 if f(n) lies between k and 2k-1. Alec has already [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4563 done this for k=1] and [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4734 partially done it for k=3].
*Search for the longest sequence of discrepancy 2 with the property that <math>x_n=x_{32n}</math> for every n. The motivation for this is to produce a fundamentally different class of examples (different because their group structure would include an element of order 5). It's not clear that it will work, since 32 is a fairly large number. However, if you've chosen <math>x_{32n}</math> then that will have some influence on several other choices, such as <math>x_{4n},x_{8n}</math> and <math>x_{16n}</math>, so maybe it will lead to something interesting. Alec [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4873 has made a start on this] and an [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4874 initial investigation] suggests that the sequence he has found does indeed have some <math>C_{10}</math>-related structure.
*Here's another experiment that should be pretty easy to program and might yield something interesting. It's to look at the how the discrepancy appears to grow when you define a sequence using a greedy algorithm. I say "a" greedy algorithm because there are various algorithms that could reasonably be described as greedy. Here are a few.

<blockquote>
1. For each n let <math>x_n</math> be chosen so as to minimize the discrepancy so far, given the choices already made for <math>x_1,\dots,x_{n-1}</math>. (If this does not uniquely determine <math>x_n</math> then choose it arbitrarily, or randomly, or according to some simple rule like always equalling 1.)
</blockquote>

<blockquote>
2. Same as 1 but with additional constraints, in the hope that these make the sequence more likely to be good. For instance, one might insist that <math>x_{2k}=x_{3k}</math> for every k. Here, when choosing <math>x_n</math> one would probably want to minimize the discrepancy up to <math>x_{n+k}</math> if <math>x_{n+1},\dots,x_{n+k}</math> had already been chosen. Another obvious constraint to try is complete multiplicativity.
</blockquote>

<blockquote>
3. A greedy algorithm of sorts, but this time trying to minimize a different parameter. The first algorithm will do this: when you pick <math>x_n</math> you look, for each factor d of n, at the partial sum along the multiples of d up to but not including n. This will give you a set A of numbers (the possible partial sums). If max(A) is greater than max(-A) then you set <math>x_n=-1</math>, if max(-A) is greater than max(A) then you let <math>x_n=1</math>, and if they are equal then you make the decision according to some rule that seems sensible. But it might be that you would end up with a slower-growing discrepancy if you regarded A as a multiset and made the decision on some other basis. For instance, you could take the sum of <math>2^k</math> over all positive elements <math>k\in A</math> (with multiplicity) and the sum of <math>2^{-k}</math> over all negative elements and choose <math>x_n</math> according to which was bigger. Although that wouldn't minimize the discrepancy at each stage, it might make the sequence better for future development because it wouldn't sacrifice the needs of an overwhelming majority to those of a few rogue elements.
</blockquote>

<blockquote>
4. A greedy algorithm to choose a good completely multiplicative low-discrepancy sequence. Now you are free only to choose the values at primes. If you have chosen the values up to but not including p, then fill in all the values that are forced by multiplicativity and then make whatever seems to be the best choice for the value at p. Again, there are several approaches that could be reasonable here. One is simply to ensure that the partial sum of the sequence up to p is as small (in modulus) as you can make it. But that would be foolish if you've already filled in the values at p+1,...,p+k. So an only slightly less greedy algorithm is to look at the effect of your choice at p on the partial sums all the way up to the next prime and choose the best value accordingly. If you do that, then at what rate do the partial sums grow? In particular, do they grow at least logarithmically?
</blockquote>

<blockquote>
The motivation for these experiments is to see whether they, or some variants, appear to lead to sublogarithmic growth. If they do, then we could start trying to prove rigorously that sublogarithmic growth is possible. I still think that a function that arises in nature and satisfies f(1124)=2 ought to be sublogarithmic.
</blockquote>

*What happens if one applies a backtracking algorithm to try to extend the following discrepancy-2 sequence, which satisfies <math>x_{2n}=-x_n</math> for every n, to a much longer discrepancy-2 sequence: + - - + - + + - - + + - + - + + - + - - + - - + + - - + + - + - - + - - + + + + - - - + + + - - + - + + - + - - + - ? This question has been answered [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4893 in the comments following the asking of the question on the blog].

* Investigate what happens if our HAPs are restricted to allow differences divisible only by 2 or 3 [and then other sets of primes including 2] - {2,3,5,7} would be interesting - is there an infinite sequence of discrepancy 2 in these simple cases - is it easy to find an infinite sequence with finite discrepancy in these cases? [for sets of odd primes, take a sequence which is 1 on odd numbers, -1 on even numbers. Including 2 is the non-trivial case]. It is possible that completely multiplicative sequences could be found for some of these cases.

* Compute the Dirichlet series <math>f(s) = \sum x_n n^{-s}</math> for some of our long low-discrepancy series, and see what this function looks like in the vicinity of <math>1</math>, and elsewhere. [http://gowers.wordpress.com/2010/01/11/the-erds-discrepancy-problem-iii/#comment-5062 Alec has now looked at this].

*Take a long sequence <math>(x_n)</math> of discrepancy 2 and try to create a new long sequence <math>(y_n)</math> subject to the constraint that <math>y_{2n}=x_n</math>. How far does one typically get before getting stuck? And how much further does one get if one uses the resulting sequence as a seed for the usual algorithm?

* ... you are welcome to add more.

Multiplicative sequences

2010-01-14T13:05:55Z

Thomas: improving clarity

Multiplicative sequences

2010-01-14T12:49:56Z

Thomas: added section on general case and some data

Short sequences statistics

2010-01-13T14:36:00Z

Thomas: added some data, and link to multiplicative data

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Click here to go back to the main experimental results page].

Here are statistics on short sequences obtained by hand or with Alec's code. Feel free to add more.

For a discrepancy C=2 there are:
* 6 sequences of length 3
* 12 sequences of length 4
* 18 sequences of length 5
* 18 sequences of length 6
* ...
* 8 436 986 sequences of length 48, of which 89 are multiplicative
* ...
* ????? sequences of length 96, of which 119 are multiplicative
* ????? ? sequences of length 192, of which 304 are multiplicative

More precise data about multiplicave sequences themselves is available [http://michaelnielsen.org/polymath1/index.php?title=Multiplicative_sequences on this page].

Multiplicative sequences

2010-01-13T14:23:05Z

Thomas: added data and link to plot

Create tables in an HTML file from an input sequence

2010-01-13T09:03:49Z

Thomas: fixed a little offset for the first table

[[The Erdős discrepancy problem|To return to the main Polymath5 page, click here]].

Here is a C++ code which you can compile with e.g. gcc under linux or cygwin.

It allows to create an HTML file displaying a sequence as well as its HAPs.

A first table contains the sequence, a second one contains it together with HAPs displayed as columns.

It takes as input a sequence in the stringent format of a file (say mysequence.txt) starting immediately with + or – and separating each element with a whitespace (put also a white space after the last element), without ever adding any carriage returns. This very format is nearly available on the wiki, just copy-paste to some file, add the spaces and suppress the non-wanted carriage returns.

Obviously you should call the output file mytable.html or something.

Feel free to add further functionalities. I'm not claiming it's fantastically written, but it works well.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
11-jan-2010: written from scratch. Really basic code...
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------
const int M=10000;
char filenameIN[50], filenameOUT[50];
ofstream myfileOUT;
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//----------------------------------------------------------
//-- main:
//
//we ask the user the name of a file containing a sequence
//and output a full (X)HTML page for use with a browser.
//
//format MUST be: starts with either + or -, each sign separated
//by a whitespace including the last one, no carriage returns,
//in particular no extra blank lines at the end.
//This very format is nearly available on the wiki, just copy-paste to
//some file and suppress the non-wanted carriage returns.
//
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, j, c,imax=0;
int datab[M];
string line, buff;
string myplus ("+");
string myminus ("-");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output TABLES filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

//fill table with zeros
for(i=0;i<M;i++){
datab[i]=0;
}

//-- read the sequence
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;

getline(myfileIN,line);

stringstream stsm(line);

while(stsm>>buff){

if(buff.compare(myplus)==0){
datab[i]=1;
//cout<<"took 1"<<endl;
}

if(buff.compare(myminus)==0){
datab[i]=-1;
//cout<<"took -1"<<endl;
}

if( datab[i] !=1 && datab[i] !=-1 ){
cout<<"problem: element number "<<i+1<<" is neither + nor -."<<endl;
abort();
}

//cout<<endl;
i+=1;

}

imax=i;
cout<<" this sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}

myfileIN.close();

//-- main processing:
//first a table containg the full sequence
//next a table showing the first 20 terms of the sequence and
//its HAPs as columns.

myfileOUT.open(filenameOUT,ios::app);

myfileOUT<<"<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Strict//EN\" ";
myfileOUT<<"\"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.tdt\"> \n\n";
myfileOUT<<"<html xmlns=\"http://www.w3.org/1999/xhtml\"> \n\n";
myfileOUT<<"<head>\n <title>a sequence and its HAPs</title>\n </head>\n";
myfileOUT<<"<body>\n";

//-- first table

myfileOUT<<" Full sequence in a squarish table:"<<endl;
myfileOUT<<"<table border=\"1\">"<<endl;

//set number of columns to sqrt(length) roughly
k=int( floor(sqrt(imax)) );

c=0;

d=imax/k;

//loop on rows
for(j=0;j<d;j++){

//construct one row, cell by cell
myfileOUT<<"<tr>"<<endl;
for(i=0;i<k;i++){

if(i==0 && j==0){
myfileOUT<<"<td bgcolor=\"black\">0</td>"<<endl;
}
else{
c+=1;
if(datab[j*k+i-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<j*k+i<<"+</td>"<<endl;
}
else{
myfileOUT<<"<td bgcolor=\"white\">"<<j*k+i<<"-</td>"<<endl;
}
}

}
myfileOUT<<"</tr>"<<endl;

}

//add final cells if last row is incomplete
if((imax-1)%k != 0){
myfileOUT<<"<tr>"<<endl;
for(u=c+1;u<=imax;u++){

if(datab[u-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<u<<"+</td>"<<endl;
}
else{
myfileOUT<<"<td bgcolor=\"white\">"<<u<<"-</td>"<<endl;
}
}
myfileOUT<<"</tr>"<<endl;

}

//finish table
myfileOUT<<"</table><br><br><br>"<<endl;

//-- now the table of HAPs as columns, first 30 elements each as rows

myfileOUT<<"Now the HAPs as columns:<br> "<<endl;
myfileOUT<<" first column is the sequence, second column is x_{2n},..."<<endl;
myfileOUT<<"<table border=\"1\">"<<endl;

//set number of columns to length/6 roughly
// (i.e. we want subsequences which have at least 6 elements)
k=int( floor(imax/6) );

//take only 30 first rows
d=30;

//loop on rows
for(j=0;j<=d;j++){

//construct one row, cell by cell
myfileOUT<<"<tr>"<<endl;
for(i=0;i<k;i++){

if(datab[(j+1)*(i+1)-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<(j+1)*(i+1)<<"+</td>"<<endl;
}

if(datab[(j+1)*(i+1)-1]==-1){
myfileOUT<<"<td bgcolor=\"white\">"<<(j+1)*(i+1)<<"-</td>"<<endl;
}
if(i*j>=M || datab[j*(i+1)]==0){
myfileOUT<<"<td bgcolor=\"grey\">.</td>"<<endl;
}
}

myfileOUT<<"</tr>"<<endl;
}

//finish table
myfileOUT<<"</table>"<<endl;

//finish HTML
myfileOUT<<"</body>\n </html>"<<endl;

//close file!
myfileOUT.close();

//exit normally
return 0;

}
</pre>

Short sequences statistics

2010-01-13T07:08:42Z

Thomas: corrected several errors

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Click here to go back to the main experimental results page].

Here are statistics on short sequences obtained with Alec's code. Feel free to add more.

For a discrepancy C=2 there are:
* 89 multiplicative sequences of length 48, out of 8 436 986 sequences of length 48
* 119 multiplicative sequences of length 96, out of ? sequences of length 96
* 304 multiplicative sequences of length 192, out of ? sequences of length 192

Short sequences statistics

2010-01-13T07:02:10Z

Thomas: added link back to exp results

[http://michaelnielsen.org/polymath1/index.php?title=Experimental_results Click here to go back to the main experimental results page].

Here are statistics on short sequences obtained with Alec's code. Feel free to add more.

Using as a seed {0, +1} it found:
* 2877 sequences of length 48
* 50404 sequences of length 96

Short sequences statistics

2010-01-13T06:59:38Z

Thomas: New page: Here are statistics on short sequences obtained with Alec's code. Feel free to add more. Using as a seed {0, +1} it found: * 2877 sequences of length 48 * 50404 sequences of length 96

Here are statistics on short sequences obtained with Alec's code. Feel free to add more.

Using as a seed {0, +1} it found:
* 2877 sequences of length 48
* 50404 sequences of length 96

Experimental results

2010-01-13T06:56:04Z

Thomas: added link to table of short sequences

[[The Erdős discrepancy problem|To return to the main Polymath5 page, click here]].

Perhaps we should have two kinds of subpages to this page: Pages about finding examples, and pages about analyzing them?

== Experimental data==
* [[The first 1124-sequence]] with discrepancy 2. ''Some more description''
* Other [[length 1124 sequences]] with discrepancy 2. ''Some more description''
* A [[sequence of length 1112]] derived from one with nice multiplicative properties.
* Some data about the problem with [[different upper and lower bound]]. Let N(a,b) be the largest N such that there is a sequence <math>x_1,\dots,x_N</math> all of whose HAP-errors are between -a and b, inclusive.
* Sequences taking values in <math>\mathbb{T}</math>:
** [[4th roots of unity]]
** [[6th roots of unity]]
* [http://thomas1111.wordpress.com/2010/01/10/tables-for-a-c10-candidate/ A sequence of length 407] with discrepancy 2 such that <math>x_n=x_{32 n}</math> for every n. [[The HAP-subsequence structure of that sequence]].
* More [[T32-invariant sequences]].
* Long [[multiplicative sequences]].
* Long sequences satisfying [[T2(x) = -x]].
* Long sequences satisfying [[T2(x) = T5(x) = -x]]
* Long sequences satisfying [[T2(x) = -T3(x)]].
* Long sequences satisfying constraints of the form [[T_m(x) = (+/-)T_n(x)]].
* Table of [[longest constrained sequences]].
* Table of [[short sequences statistics]].

==Source code==

* [[Convert raw input string into CSV table]]
* [[Create tables in an HTML file from an input sequence]]
* [[Verify the bounded discrepancy of an input sequence]]
* [[Depth-first search]]

==Wish list==

* Find long/longest quasi-multiplicative sequences with some fixed group G, function <math>G\to \{-1,1\}</math> and maximal discrepancy C
** <math>G=C_6</math> and the function that sends 0,1 and 2 to 1 (because this seems to be a good choice)
* Do a "Mark-Bennet-style analysis" of one of the new 1124-sequences. [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4827] Also [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4842 done] (by Mark Bennet).
*. Take a moderately large k and search for the longest sequence of discrepancy 2 that's constructed as follows. First, pick a completely multiplicative function f to the group <math>C_{2k}</math>. Then set <math>x_n</math> to be 1 if f(n) lies between 0 and k-1, and -1 if f(n) lies between k and 2k-1. Alec has already [http://gowers.wordpress.com/2009/12/17/erdoss-discrepancy-problem/#comment-4563 done this for k=1] and [http://gowers.wordpress.com/2010/01/06/erdss-discrepancy-problem-as-a-forthcoming-polymath-project/#comment-4734 partially done it for k=3].
*Search for the longest sequence of discrepancy 2 with the property that <math>x_n=x_{32n}</math> for every n. The motivation for this is to produce a fundamentally different class of examples (different because their group structure would include an element of order 5). It's not clear that it will work, since 32 is a fairly large number. However, if you've chosen <math>x_{32n}</math> then that will have some influence on several other choices, such as <math>x_{4n},x_{8n}</math> and <math>x_{16n}</math>, so maybe it will lead to something interesting. Alec [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4873 has made a start on this] and an [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4874 initial investigation] suggests that the sequence he has found does indeed have some <math>C_{10}</math>-related structure.
*Here's another experiment that should be pretty easy to program and might yield something interesting. It's to look at the how the discrepancy appears to grow when you define a sequence using a greedy algorithm. I say "a" greedy algorithm because there are various algorithms that could reasonably be described as greedy. Here are a few.

<blockquote>
1. For each n let <math>x_n</math> be chosen so as to minimize the discrepancy so far, given the choices already made for <math>x_1,\dots,x_{n-1}</math>. (If this does not uniquely determine <math>x_n</math> then choose it arbitrarily, or randomly, or according to some simple rule like always equalling 1.)
</blockquote>

<blockquote>
2. Same as 1 but with additional constraints, in the hope that these make the sequence more likely to be good. For instance, one might insist that <math>x_{2k}=x_{3k}</math> for every k. Here, when choosing <math>x_n</math> one would probably want to minimize the discrepancy up to <math>x_{n+k}</math> if <math>x_{n+1},\dots,x_{n+k}</math> had already been chosen. Another obvious constraint to try is complete multiplicativity.
</blockquote>

<blockquote>
3. A greedy algorithm of sorts, but this time trying to minimize a different parameter. The first algorithm will do this: when you pick <math>x_n</math> you look, for each factor d of n, at the partial sum along the multiples of d up to but not including n. This will give you a set A of numbers (the possible partial sums). If max(A) is greater than max(-A) then you set <math>x_n=-1</math>, if max(-A) is greater than max(A) then you let <math>x_n=1</math>, and if they are equal then you make the decision according to some rule that seems sensible. But it might be that you would end up with a slower-growing discrepancy if you regarded A as a multiset and made the decision on some other basis. For instance, you could take the sum of <math>2^k</math> over all positive elements <math>k\in A</math> (with multiplicity) and the sum of <math>2^{-k}</math> over all negative elements and choose <math>x_n</math> according to which was bigger. Although that wouldn't minimize the discrepancy at each stage, it might make the sequence better for future development because it wouldn't sacrifice the needs of an overwhelming majority to those of a few rogue elements.
</blockquote>

<blockquote>
4. A greedy algorithm to choose a good completely multiplicative low-discrepancy sequence. Now you are free only to choose the values at primes. If you have chosen the values up to but not including p, then fill in all the values that are forced by multiplicativity and then make whatever seems to be the best choice for the value at p. Again, there are several approaches that could be reasonable here. One is simply to ensure that the partial sum of the sequence up to p is as small (in modulus) as you can make it. But that would be foolish if you've already filled in the values at p+1,...,p+k. So an only slightly less greedy algorithm is to look at the effect of your choice at p on the partial sums all the way up to the next prime and choose the best value accordingly. If you do that, then at what rate do the partial sums grow? In particular, do they grow at least logarithmically?
</blockquote>

<blockquote>
The motivation for these experiments is to see whether they, or some variants, appear to lead to sublogarithmic growth. If they do, then we could start trying to prove rigorously that sublogarithmic growth is possible. I still think that a function that arises in nature and satisfies f(1124)=2 ought to be sublogarithmic.
</blockquote>

*What happens if one applies a backtracking algorithm to try to extend the following discrepancy-2 sequence, which satisfies <math>x_{2n}=-x_n</math> for every n, to a much longer discrepancy-2 sequence: + - - + - + + - - + + - + - + + - + - - + - - + + - - + + - + - - + - - + + + + - - - + + + - - + - + + - + - - + - ? This question has been answered [http://gowers.wordpress.com/2010/01/09/erds-discrepancy-problem-continued/#comment-4893 in the comments following the asking of the question on the blog].

* Investigate what happens if our HAPs are restricted to allow differences divisible only by 2 or 3 [and then other sets of primes including 2] - {2,3,5,7} would be interesting - is there an infinite sequence of discrepancy 2 in these simple cases - is it easy to find an infinite sequence with finite discrepancy in these cases? [for sets of odd primes, take a sequence which is 1 on odd numbers, -1 on even numbers. Including 2 is the non-trivial case]. It is possible that completely multiplicative sequences could be found for some of these cases.

* Compute the Dirichlet series <math>f(s) = \sum x_n n^{-s}</math> for some of our long low-discrepancy series, and see what this function looks like in the vicinity of <math>1</math>, and elsewhere.

* ... you are welcome to add more.

Create tables in an HTML file from an input sequence

2010-01-11T17:43:18Z

Thomas: further commenting out

[[The Erdős discrepancy problem|To return to the main Polymath5 page, click here]].

Here is a C++ code which you can compile with e.g. gcc under linux or cygwin.

It allows to create an HTML file displaying a sequence as well as its HAPs.

A first table contains the sequence, a second one contains it together with HAPs displayed as columns.

It takes as input a sequence in the stringent format of a file (say mysequence.txt) starting immediately with + or – and separating each element with a whitespace (put also a white space after the last element), without ever adding any carriage returns. This very format is nearly available on the wiki, just copy-paste to some file, add the spaces and suppress the non-wanted carriage returns.

Obviously you should call the output file mytable.html or something.

Feel free to add further functionalities. I'm not claiming it's fantastically written, but it works well.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
11-jan-2010: written from scratch. Really basic code...
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------
const int M=10000;
char filenameIN[50], filenameOUT[50];
ofstream myfileOUT;
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//----------------------------------------------------------
//-- main:
//
//we ask the user the name of a file containing a sequence
//and output a full (X)HTML page for use with a browser.
//
//format MUST be: starts with either + or -, each sign separated
//by a whitespace including the last one, no carriage returns,
//in particular no extra blank lines at the end.
//This very format is nearly available on the wiki, just copy-paste to
//some file and suppress the non-wanted carriage returns.
//
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, j, c,imax=0;
int datab[M];
string line, buff;
string myplus ("+");
string myminus ("-");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output TABLES filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

//fill table with zeros
for(i=0;i<M;i++){
datab[i]=0;
}

//-- read the sequence
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;

getline(myfileIN,line);

stringstream stsm(line);

while(stsm>>buff){

if(buff.compare(myplus)==0){
datab[i]=1;
//cout<<"took 1"<<endl;
}

if(buff.compare(myminus)==0){
datab[i]=-1;
//cout<<"took -1"<<endl;
}

if( datab[i] !=1 && datab[i] !=-1 ){
cout<<"problem: element number "<<i+1<<" is neither + nor -."<<endl;
abort();
}

//cout<<endl;
i+=1;

}

imax=i;
cout<<" this sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}

myfileIN.close();

//-- main processing:
//first a table containg the full sequence
//next a table showing the first 20 terms of the sequence and
//its HAPs as columns.

myfileOUT.open(filenameOUT,ios::app);

myfileOUT<<"<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Strict//EN\" ";
myfileOUT<<"\"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.tdt\"> \n\n";
myfileOUT<<"<html xmlns=\"http://www.w3.org/1999/xhtml\"> \n\n";
myfileOUT<<"<head>\n <title>a sequence and its HAPs</title>\n </head>\n";
myfileOUT<<"<body>\n";

//-- first table

myfileOUT<<" Full sequence in a squarish table:"<<endl;
myfileOUT<<"<table border=\"1\">"<<endl;

//set number of columns to sqrt(length) roughly
k=int( floor(sqrt(imax)) );

c=0;

d=imax/k;

//loop on rows
for(j=0;j<d;j++){

//construct one row, cell by cell
myfileOUT<<"<tr>"<<endl;
for(i=0;i<k;i++){

if(i==0 && j==0){
myfileOUT<<"<td bgcolor=\"black\">0</td>"<<endl;
}
else{
c+=1;
if(datab[j*k+i-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<j*k+i<<"+</td>"<<endl;
}
else{
myfileOUT<<"<td bgcolor=\"white\">"<<j*k+i<<"-</td>"<<endl;
}
}

}
myfileOUT<<"</tr>"<<endl;

}

//add final cells if last row is incomplete
if(imax%k != 0){
myfileOUT<<"<tr>"<<endl;
for(u=c+1;u<=imax;u++){

if(datab[u-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<u<<"+</td>"<<endl;
}
else{
myfileOUT<<"<td bgcolor=\"white\">"<<u<<"-</td>"<<endl;
}
}
myfileOUT<<"</tr>"<<endl;

}

//finish table
myfileOUT<<"</table><br><br><br>"<<endl;

//-- now the table of HAPs as columns, first 30 elements each as rows

myfileOUT<<"Now the HAPs as columns:<br> "<<endl;
myfileOUT<<" first column is the sequence, second column is x_{2n},..."<<endl;
myfileOUT<<"<table border=\"1\">"<<endl;

//set number of columns to length/6 roughly
// (i.e. we want subsequences which have at least 6 elements)
k=int( floor(imax/6) );

//take only 30 first rows
d=30;

//loop on rows
for(j=0;j<=d;j++){

//construct one row, cell by cell
myfileOUT<<"<tr>"<<endl;
for(i=0;i<k;i++){

if(datab[(j+1)*(i+1)-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<(j+1)*(i+1)<<"+</td>"<<endl;
}

if(datab[(j+1)*(i+1)-1]==-1){
myfileOUT<<"<td bgcolor=\"white\">"<<(j+1)*(i+1)<<"-</td>"<<endl;
}
if(i*j>=M || datab[j*(i+1)]==0){
myfileOUT<<"<td bgcolor=\"grey\">.</td>"<<endl;
}
}

myfileOUT<<"</tr>"<<endl;
}

//finish table
myfileOUT<<"</table>"<<endl;

//finish HTML
myfileOUT<<"</body>\n </html>"<<endl;

//close file!
myfileOUT.close();

//exit normally
return 0;

}
</pre>

Create tables in an HTML file from an input sequence

2010-01-11T17:14:10Z

Thomas: added link back to polymath5 page

[[The Erdős discrepancy problem|To return to the main Polymath5 page, click here]].

Here is a C++ code which you can compile with e.g. gcc under linux or cygwin.

It allows to create an HTML file displaying a sequence as well as its HAPs.

A first table contains the sequence, a second one contains it together with HAPs displayed as columns.

It takes as input a sequence in the stringent format of a file (say mysequence.txt) starting immediately with + or – and separating each element with a whitespace (put also a white space after the last element), without ever adding any carriage returns. This very format is nearly available on the wiki, just copy-paste to some file and suppress the non-wanted carriage returns.

Obviously you should call the output file mytable.html or something.

Feel free to add further functionalities. I'm not claiming it's fantastically written, but it works well.

<pre>
/*----------------------------------------------------------
author: Thomas Sauvaget
licence: public domain.
files: just this one.
------------------------------------------------------------
modification record:
11-jan-2010: written from scratch. Really basic code...
----------------------------------------------------------*/

//---------------------------------------------------------
//-- libraries & preprocessor
//---------------------------------------------------------
#include <iostream> //for imput-output with terminal
#include <cmath> //small math library
#include <algorithm> //useful for permutations and the likes
#include <vector>
#include <fstream> //for imput-output with files
#include <iomanip> //for precision control when outputing
#include <stdlib.h>
#include <string>
#include <sstream>
//#include <assert> //for end of file command

//---------------------------------------------------------
//-- namespace (standard)
//---------------------------------------------------------
using namespace std;

//---------------------------------------------------------
//-- global variables
//---------------------------------------------------------
const int M=10000;
char filenameIN[50], filenameOUT[50];
ofstream myfileOUT;
ifstream myfileIN;

//---------------------------------------------------------
//-- useful handmade functions
//---------------------------------------------------------

//---------
//getintval
//converts string containing a single integer to int
//---------
int getintval(string strconv){
int intret;

intret=atoi(strconv.c_str());

return(intret);
}

//----------------------------------------------------------
//-- main:
//
//we ask the user the name of a file containing a sequence
//and output a full (X)HTML page for use with a browser.
//
//format MUST be: starts with either + or -, each sign separated
//by a whitespace including the last one, no carriage returns,
//in particular no extra blank lines at the end.
//This very format is nearly available on the wiki, just copy-paste to
//some file and suppress the non-wanted carriage returns.
//
//----------------------------------------------------------
int main(int argc, char *argv[]){

int i, k, d, u, v, j, c,imax=0;
int datab[M];
string line, buff;
string myplus ("+");
string myminus ("-");

//-- textlike user interface
cout<<endl;
cout<<" give input SEQUENCE filename:"<<endl;
cin>>filenameIN;
cout<<endl;

cout<<endl;
cout<<" give output TABLES filename:"<<endl;
cin>>filenameOUT;
cout<<endl;

//-- file creation
myfileOUT.open(filenameOUT,ios::out);
myfileOUT.close();

//fill table with zeros
for(i=0;i<M;i++){
datab[i]=0;
}

//-- read the sequence
myfileIN.open(filenameIN,ios::in);

if(myfileIN.is_open()){

i=0;

getline(myfileIN,line);

stringstream stsm(line);

while(stsm>>buff){

if(buff.compare(myplus)==0){
datab[i]=1;
//cout<<"took 1"<<endl;
}

if(buff.compare(myminus)==0){
datab[i]=-1;
//cout<<"took -1"<<endl;
}

if( datab[i] !=1 && datab[i] !=-1 ){
cout<<"problem: element number "<<i+1<<" is neither + nor -."<<endl;
abort();
}

cout<<endl;
i+=1;

}

imax=i;
cout<<" this sequence contains "<<imax<<" elements."<<endl;
cout<<endl;

}
else{
cout<<"there is no file with this very name"<<endl;
cout<<" (don't forget extension like .txt or .dat in particular)"<<endl;
abort();
}

myfileIN.close();

//-- main processing:
//first a table containg the full sequence
//next a table showing the first 20 terms of the sequence and
//its HAPs as columns.

myfileOUT.open(filenameOUT,ios::app);

myfileOUT<<"<!DOCTYPE html PUBLIC \"-//W3C//DTD XHTML 1.0 Strict//EN\" ";
myfileOUT<<"\"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.tdt\"> \n\n";
myfileOUT<<"<html xmlns=\"http://www.w3.org/1999/xhtml\"> \n\n";
myfileOUT<<"<head>\n <title>a sequence and its HAPs</title>\n </head>\n";
myfileOUT<<"<body>\n";

//-- first table

myfileOUT<<" Full sequence in a squarish table:"<<endl;
myfileOUT<<"<table border=\"1\">"<<endl;

//set number of columns to sqrt(length) roughly
k=int( floor(sqrt(imax)) );

c=0;

d=imax/k;

//loop on rows
for(j=0;j<d;j++){

//construct one row, cell by cell
myfileOUT<<"<tr>"<<endl;
for(i=0;i<k;i++){

if(i==0 && j==0){
myfileOUT<<"<td bgcolor=\"black\">0</td>"<<endl;
}
else{
c+=1;
if(datab[j*k+i-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<j*k+i<<"+</td>"<<endl;
}
else{
myfileOUT<<"<td bgcolor=\"white\">"<<j*k+i<<"-</td>"<<endl;
}
}

}
myfileOUT<<"</tr>"<<endl;

}

//add final cells if last row is incomplete
if(imax%k != 0){
myfileOUT<<"<tr>"<<endl;
for(u=c+1;u<=imax;u++){

if(datab[u-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<u<<"+</td>"<<endl;
}
else{
myfileOUT<<"<td bgcolor=\"white\">"<<u<<"-</td>"<<endl;
}
}
myfileOUT<<"</tr>"<<endl;

}

//finish table
myfileOUT<<"</table><br><br><br>"<<endl;

//-- now the table of HAPs as columns, first 30 elements each as rows

myfileOUT<<"Now the HAPs as columns:<br> "<<endl;
myfileOUT<<" first column is the sequence, second column is x_{2n},..."<<endl;
myfileOUT<<"<table border=\"1\">"<<endl;

//set number of columns to length/6 roughly
// (i.e. we want subsequences which have at least 6 elements)
k=int( floor(imax/6) );

//take only 30 first rows
d=30;

//loop on rows
for(j=0;j<=d;j++){

//construct one row, cell by cell
myfileOUT<<"<tr>"<<endl;
for(i=0;i<k;i++){

if(datab[(j+1)*(i+1)-1]==1){
myfileOUT<<"<td bgcolor=\"cyan\">"<<(j+1)*(i+1)<<"+</td>"<<endl;
}

if(datab[(j+1)*(i+1)-1]==-1){
myfileOUT<<"<td bgcolor=\"white\">"<<(j+1)*(i+1)<<"-</td>"<<endl;
}
if(i*j>=M || datab[j*(i+1)]==0){
myfileOUT<<"<td bgcolor=\"grey\">.</td>"<<endl;
}
}

myfileOUT<<"</tr>"<<endl;
}

//finish table
myfileOUT<<"</table>"<<endl;

//finish HTML
myfileOUT<<"</body>\n </html>"<<endl;

//close file!
myfileOUT.close();

//exit normally
return 0;

}
</pre>