Computer proof that completely multiplicative sequences have discrepancy greater than 2

From Polymath1Wiki
Revision as of 09:59, 3 February 2010 by Thomas (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

The following C code implements 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]