This is a sub-page for the Polymath8 project "bounded gaps between primes".
- [math]\displaystyle{ ~k_0~ }[/math] is a quantity such that every admissible [math]\displaystyle{ ~k_0 }[/math]-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible.
- [math]\displaystyle{ \text{MPZ}^{(i)}[\varpi,\delta] }[/math] holds for some combinations of [math]\displaystyle{ c_\varpi, c_\delta }[/math], and [math]\displaystyle{ ~i~ }[/math] values, where [math]\displaystyle{ i \ge 1 }[/math] means [math]\displaystyle{ ~i }[/math]-tuply densely divisible, [math]\displaystyle{ c_\varpi \gt 0 }[/math] and [math]\displaystyle{ ~c_\delta \gt 0~ }[/math] are constants in the constraint on [math]\displaystyle{ \varpi }[/math] and [math]\displaystyle{ ~\delta~ }[/math], such that [math]\displaystyle{ c_{\varpi}\varpi+c_{\delta}\delta\lt 1 }[/math].
Benchmarks
Demonstration results at [math]\displaystyle{ k_0 = k_0^{opt} }[/math] for the best currently known instances of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math] values.
| Instance
|
[math]\displaystyle{ k_0^{*} }[/math]
|
[math]\displaystyle{ ~k_0~ }[/math]
|
Parameters
|
Error Terms
|
Comment
|
| [math]\displaystyle{ c_{\varpi} }[/math] |
[math]\displaystyle{ ~c_{\delta}~ }[/math] |
[math]\displaystyle{ ~i~ }[/math] |
[math]\displaystyle{ \varpi }[/math] |
[math]\displaystyle{ ~\delta~ }[/math] |
[math]\displaystyle{ ~\delta'~ }[/math] |
[math]\displaystyle{ ~A~ }[/math] |
[math]\displaystyle{ ~\kappa_1~ }[/math] |
[math]\displaystyle{ ~\kappa_2~ }[/math] |
[math]\displaystyle{ ~\kappa_3~ }[/math]
|
| 168
|
48
|
2
|
1781
|
1783
|
5.950000E-03
|
1E-05
|
1/300
|
800
|
6.662E-07
|
5.209E-09
|
8.340E-47
|
Without Deligne's theorem
|
| 600/7
|
180/7
|
4
|
630
|
632
|
1.163666E-02
|
1E-04
|
1/105
|
200
|
6.445E-06
|
1.752E-08
|
7.018E-08
|
With Deligne's theorem
|
Optimal results at [math]\displaystyle{ k_0 = k_0^{opt} }[/math] for some instances of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math] values.
| Instance
|
[math]\displaystyle{ k_0^{*} }[/math]
|
[math]\displaystyle{ ~k_0~ }[/math]
|
Parameters
|
Error Terms
|
Objective
|
| [math]\displaystyle{ c_{\varpi} }[/math] |
[math]\displaystyle{ ~c_{\delta}~ }[/math] |
[math]\displaystyle{ ~i~ }[/math] |
[math]\displaystyle{ \varpi }[/math] |
[math]\displaystyle{ ~\delta~ }[/math] |
[math]\displaystyle{ ~\delta'~ }[/math] |
[math]\displaystyle{ ~A~ }[/math] |
[math]\displaystyle{ ~\kappa_1~ }[/math] |
[math]\displaystyle{ ~\kappa_2~ }[/math] |
[math]\displaystyle{ ~\kappa_3~ }[/math]
|
| 348
|
68
|
1
|
5446
|
5447
|
2.8733351E-03
|
1.1672627E-06
|
1.4961657E-03
|
2559.258877
|
5.59E-09
|
1.50E-12
|
6.02E-11
|
-1.1882E-06
|
| 168
|
48
|
2
|
1781
|
1783
|
5.9495534E-03
|
9.8965035E-06
|
3.7117059E-03
|
757.8242621
|
1.58E-07
|
3.24E-10
|
3.65E-09
|
-5.9684E-06
|
| 148
|
33
|
1
|
1465
|
1466
|
6.7542244E-03
|
1.1357314E-05
|
4.7101572E-03
|
626.6135921
|
8.79E-08
|
8.57E-11
|
3.63E-09
|
-2.2867E-06
|
| 140
|
32
|
1
|
1345
|
1346
|
7.1398444E-03
|
1.3180858E-05
|
5.0540952E-03
|
577.7849932
|
1.10E-07
|
1.22E-10
|
4.75E-09
|
-6.7812E-06
|
| 116
|
30
|
1
|
1006
|
1007
|
8.6150244E-03
|
2.1905745E-05
|
6.4310210E-03
|
408.9674914
|
2.29E-07
|
3.76E-10
|
1.20E-08
|
-6.2561E-06
|
| 108
|
30
|
1
|
901
|
902
|
9.2518776E-03
|
2.6573843E-05
|
7.0318847E-03
|
359.6376563
|
3.08E-07
|
6.00E-10
|
1.76E-08
|
-1.0924E-05
|
| 280/3
|
80/3
|
2
|
719
|
720
|
1.0699851E-02
|
5.0521044E-05
|
8.0398983E-03
|
260.2624368
|
1.04E-06
|
4.98E-09
|
4.33E-08
|
-5.5687E-06
|
| 600/7
|
180/7
|
4
|
630
|
632
|
1.1639206E-02
|
9.1536798E-05
|
8.3866560E-03
|
194.5246551
|
3.01E-06
|
3.40E-08
|
9.89E-08
|
-5.0940E-06
|
"Failure" results at [math]\displaystyle{ k_0=(k_0^{opt}-1) }[/math] for some instances of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math] values.
| Instance
|
[math]\displaystyle{ k_0^{*} }[/math]
|
[math]\displaystyle{ ~k_0~ }[/math]
|
Parameters
|
Error Terms
|
Objective
|
| [math]\displaystyle{ c_{\varpi} }[/math] |
[math]\displaystyle{ ~c_{\delta}~ }[/math] |
[math]\displaystyle{ ~i~ }[/math] |
[math]\displaystyle{ \varpi }[/math] |
[math]\displaystyle{ ~\delta~ }[/math] |
[math]\displaystyle{ ~\delta'~ }[/math] |
[math]\displaystyle{ ~A~ }[/math] |
[math]\displaystyle{ ~\kappa_1~ }[/math] |
[math]\displaystyle{ ~\kappa_2~ }[/math] |
[math]\displaystyle{ ~\kappa_3~ }[/math]
|
| 348
|
68
|
1
|
5446
|
5446
|
2.8733354E-03
|
1.1660200E-06
|
1.4880084E-03
|
2552.313151
|
6.16E-09
|
1.81E-12
|
1.00E-10
|
1.7550E-07
|
| 168
|
48
|
2
|
1781
|
1782
|
5.9495511E-03
|
9.9043741E-06
|
3.7130742E-03
|
757.3673135
|
1.59E-07
|
3.26E-10
|
3.21E-09
|
2.5064E-06
|
| 148
|
33
|
1
|
1465
|
1465
|
6.7542239E-03
|
1.1359571E-05
|
4.7002144E-03
|
625.1479808
|
9.16E-08
|
9.27E-11
|
3.28E-09
|
9.3639E-06
|
| 140
|
32
|
1
|
1345
|
1345
|
7.1398419E-03
|
1.3191801E-05
|
5.0550954E-03
|
567.8210511
|
1.11E-07
|
1.24E-10
|
4.65E-10
|
6.6029E-06
|
| 116
|
30
|
1
|
1006
|
1006
|
8.6150194E-03
|
2.1925014E-05
|
6.4287825E-03
|
408.5511082
|
2.33E-07
|
3.89E-10
|
1.24E-08
|
1.5183E-05
|
| 108
|
30
|
1
|
901
|
901
|
9.2518661E-03
|
2.6615167E-05
|
7.0404135E-03
|
359.5845846
|
3.08E-07
|
5.97E-10
|
1.68E-08
|
1.4703E-05
|
| 280/3
|
80/3
|
2
|
719
|
719
|
1.0699822E-02
|
5.0626919E-05
|
8.0520479E-03
|
259.8370595
|
1.04E-06
|
4.96E-09
|
4.46E-08
|
3.1365E-05
|
| 600/7
|
180/7
|
4
|
630
|
631
|
1.1639134E-02
|
9.1775130E-05
|
8.3989836E-03
|
193.9881059
|
3.02E-06
|
3.40E-08
|
1.00E-07
|
4.0614E-05
|
Conservative "Failure" results at [math]\displaystyle{ k_0=(k_0^{opt}-1) }[/math] for some instances of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math] values.
| Instance
|
[math]\displaystyle{ k_0^{*} }[/math]
|
[math]\displaystyle{ ~k_0~ }[/math]
|
Parameters
|
Error Terms
|
Objective
|
| [math]\displaystyle{ c_{\varpi} }[/math] |
[math]\displaystyle{ ~c_{\delta}~ }[/math] |
[math]\displaystyle{ ~i~ }[/math] |
[math]\displaystyle{ \varpi }[/math] |
[math]\displaystyle{ ~\delta~ }[/math] |
[math]\displaystyle{ ~\delta'~ }[/math] |
[math]\displaystyle{ ~A~ }[/math] |
[math]\displaystyle{ ~\kappa_1~ }[/math] |
[math]\displaystyle{ ~\kappa_2~ }[/math] |
[math]\displaystyle{ ~\kappa_3~ }[/math]
|
| 168
|
48
|
2
|
1781
|
1782
|
5.9501100E-03
|
7.9483333E-06
|
1.9082658E-03
|
777.7015422
|
1.68E-04
|
2.02E-04
|
9.81E-06
|
7.6073E-04
|
| 600/7
|
180/7
|
4
|
630
|
631
|
1.1648112E-02
|
6.1848056E-05
|
4.2588144E-03
|
222.5549310
|
9.33E-04
|
1.79E-03
|
1.02E-04
|
5.6566E-03
|
Lower Bounds
For each [math]\displaystyle{ ~c_\varpi }[/math], a theoretical lower bound of [math]\displaystyle{ ~k_0 }[/math], called [math]\displaystyle{ k_0^* }[/math], can be obtained by assuming that all error terms [math]\displaystyle{ ~\kappa_1 }[/math], [math]\displaystyle{ ~\kappa_2 }[/math], and [math]\displaystyle{ ~\kappa_3 }[/math] could be completely ignored. This table gives the computational results of [math]\displaystyle{ k_0^* }[/math] for [math]\displaystyle{ c_\varpi \lt 87 }[/math].
| [math]\displaystyle{ ~c_\varpi~ }[/math] |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9
|
| 80
|
566
|
577
|
588
|
599
|
611
|
622
|
633
|
-
|
-
|
-
|
| 70
|
460
|
470
|
481
|
491
|
502
|
512
|
523
|
533
|
544
|
555
|
| 60
|
362
|
372
|
381
|
391
|
400
|
410
|
420
|
430
|
440
|
450
|
| 50
|
273
|
281
|
290
|
299
|
307
|
316
|
325
|
334
|
343
|
353
|
| 40
|
193
|
200
|
208
|
216
|
223
|
231
|
239
|
248
|
256
|
264
|
| 30
|
123
|
129
|
136
|
143
|
149
|
156
|
163
|
171
|
178
|
185
|
| 20
|
65
|
70
|
76
|
81
|
87
|
92
|
98
|
104
|
110
|
117
|
| 10
|
22
|
26
|
30
|
33
|
38
|
42
|
46
|
51
|
55
|
60
|
| 00
|
-
|
-
|
-
|
-
|
6
|
8
|
10
|
13
|
16
|
19
|
