Finding optimal k0 values

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

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	2.8733352E-03	1.1670730E-06	1.4955362E-03	2559.258877	5.63E-09	1.52E-12	8.54E-11	-1.1881E-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	8.6150249E-03	2.1903801E-05	6.4285376E-03	408.9674914	2.30E-07	3.80E-10	1.17E-08	-6.2560E-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	1.1666317E-02	1.1659847E-06	1.4881806E-03	2558.927043	6.15E-09	1.81E-12	1.34E-10	1.7560E-07
168	48	2	1781	1782	1.1663695E-02	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	1.1663259E-02	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	1.1662709E-02	1.3191723E-05	5.0558681E-03	568.1690873	1.11E-07	1.23E-10	4.94E-10	6.6030E-06
116	30	1	1006	1006	1.1660089E-02	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	1.1658682E-02	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	1.1651479E-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