Finding optimal k0 values

This is a sub-page for the Polymath8 project "bounded gaps between primes".

• [math]\displaystyle{ ~k_0~ }[/math] ([math]\displaystyle{ ~k_0 \ge 2, k_0 \in \Z~ }[/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. It is then used for finding narrow admissible tuples.

• [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].

Optimization Model

For a given set of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math], the basic optimization model is defined as:

[math]\displaystyle{ \text{minimize}~~k_0~ }[/math]

[math]\displaystyle{ \text{subject to:}~ }[/math]

[math]\displaystyle{ 2(\kappa_1+\kappa_2+\kappa_3) \lt \left(1- \frac{j_{k_0-2}^2}{k_0(k_0-1)(1 + 4 \varpi)}\right) }[/math]

[math]\displaystyle{ c_\varpi \varpi + c_\delta \delta \lt 1, }[/math]

[math]\displaystyle{ 0 \lt \varpi \lt 1/4, }[/math]

[math]\displaystyle{ 0 \lt \delta \leq \delta' \lt \frac{1}{4} + \varpi, }[/math]

[math]\displaystyle{ A \ge 0, }[/math]

[math]\displaystyle{ \text{where}~ }[/math]

[math]\displaystyle{ \kappa_1 := \int_{\theta}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t} }[/math]

[math]\displaystyle{ \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1} \frac{dt}{t} }[/math]

[math]\displaystyle{ \kappa_3 := \tilde \theta \frac{J_{k_0-2}(\sqrt{\tilde \theta} j_{k_0-2})^2 - J_{k_0-3}(\sqrt{\tilde \theta} j_{k_0-2}) J_{k_0-1}(\sqrt{\tilde \theta} j_{k_0-2})}{ J_{k_0-3}(j_{k_0-2})^2 } \exp( A + (k_0-1) \int_{\tilde \delta}^\theta e^{-(A+2\alpha)t} \frac{dt}{t} ) }[/math]

[math]\displaystyle{ \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)} }[/math]

[math]\displaystyle{ \theta := \frac{\delta'}{1/4 + \varpi} }[/math]

[math]\displaystyle{ \tilde \theta := \frac{(i\delta' - \delta)/2 + \varpi}{1/4 + \varpi} }[/math]

[math]\displaystyle{ \tilde \delta := \frac{\delta}{1/4 + \varpi} }[/math]

and [math]\displaystyle{ \varpi, \delta, \delta', A }[/math] are parameters to be optimized.

More details are described in the page on Dickson-Hardy-Littlewood theorems.

Benchmarks

The following table provides the results with clean parameter values for the best currently known instances of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math] values, respectively without and with using Deligne's theorem.

Clean 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

The following table provides the results with minimized parameter values at [math]\displaystyle{ k_0 = k_0^{opt} }[/math] for some instances of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math] values.

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

The following table provides the "unsatisfied" results with minimized parameter values at [math]\displaystyle{ k_0=(k_0^{opt}-1) }[/math] for some instances of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math] values. Note that the results of these [math]\displaystyle{ k_0 }[/math] values do not satisfy all constraints, as indicated from the positive objective values.

"Unsatisfied" optimal 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

The following table provides the results with minimized parameter values at [math]\displaystyle{ k_0=(k_0^{opt}-1) }[/math] for some instances of [math]\displaystyle{ c_\varpi, c_\delta, i }[/math] values, with an additional condition that [math]\displaystyle{ \left(1- \frac{j_{k_0-2}^2}{k_0(k_0-1)(1 + 4 \varpi)}\right) \ge 0 }[/math]. Note that these [math]\displaystyle{ k_0 }[/math] values are more unsatisfied, as indicated from the larger positive objective values that that in the previous table.

More conservative "unsatisfied" 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

Code and data

• Code for finding optimal [math]\displaystyle{ k_0 }[/math] values: Java, bash, and maple are needed.