# Difference between revisions of "Finding optimal k0 values"

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

• $~k_0~$ ($~k_0 \ge 2, k_0 \in \Z~$) is a quantity such that every admissible $~k_0$-tuple has infinitely many translates which each contain at least two primes. It is then used for finding narrow admissible tuples.
• $\text{MPZ}^{(i)}[\varpi,\delta]$ holds for some combinations of $c_\varpi, c_\delta$, and $~i~$ values, where $i \ge 1$ means $~i$-tuply densely divisible, $c_\varpi \gt 0$ and $~c_\delta \gt 0~$ are constants in the constraint on $\varpi$ and $~\delta~$, such that $c_{\varpi}\varpi+c_{\delta}\delta\lt1$.

## Optimization Model

For a given set of $c_\varpi, c_\delta, i$, the basic optimization model is defined as:

$\text{minimize}~~k_0~$

$\text{subject to:}~$

$2(\kappa_1+\kappa_2+\kappa_3) \lt \left(1- \frac{j_{k_0-2}^2}{k_0(k_0-1)(1 + 4 \varpi)}\right)$
$c_\varpi \varpi + c_\delta \delta \lt 1,$
$0 \lt \varpi \lt1/4,$
$0 \lt \delta \leq \delta' \lt \frac{1}{4} + \varpi,$
$A \ge 0,$

$\text{where}~$

$\kappa_1 := \int_{\theta}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t}$
$\kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1} \frac{dt}{t}$
$\kappa_3 := \tilde \theta \frac{J_{k_0-2}(\sqrt{\tilde \theta} j_{k_0-2})^2 - J_{k_0-3}(\sqrt{\tilde \theta} j_{k_0-2}) J_{k_0-1}(\sqrt{\tilde \theta} j_{k_0-2})}{ J_{k_0-3}(j_{k_0-2})^2 } \exp( A + (k_0-1) \int_{\tilde \delta}^\theta e^{-(A+2\alpha)t} \frac{dt}{t} )$
$\alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}$
$\theta := \frac{\delta'}{1/4 + \varpi}$
$\tilde \theta := \frac{(i\delta' - \delta)/2 + \varpi}{1/4 + \varpi}$
$\tilde \delta := \frac{\delta}{1/4 + \varpi}$

and $\varpi, \delta, \delta', A$ 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 $c_\varpi, c_\delta, i$ values, respectively without and with using Deligne's theorem.

Clean results at $k_0 = k_0^{opt}$ for the best currently known instances of $c_\varpi, c_\delta, i$ values.
Instance $k_0^{*}$ $~k_0~$ Parameters Error Terms Comment
$c_{\varpi}$ $~c_{\delta}~$ $~i~$ $\varpi$ $~\delta~$ $~\delta'~$ $~A~$ $~\kappa_1~$ $~\kappa_2~$ $~\kappa_3~$
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 $k_0 = k_0^{opt}$ for some instances of $c_\varpi, c_\delta, i$ values.

Optimal results at $k_0 = k_0^{opt}$ for some instances of $c_\varpi, c_\delta, i$ values.
Instance $k_0^{*}$ $~k_0~$ Parameters Error Terms Objective
$c_{\varpi}$ $~c_{\delta}~$ $~i~$ $\varpi$ $~\delta~$ $~\delta'~$ $~A~$ $~\kappa_1~$ $~\kappa_2~$ $~\kappa_3~$
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 $k_0=(k_0^{opt}-1)$ for some instances of $c_\varpi, c_\delta, i$ values. Note that the results of these $k_0$ values do not satisfy all constraints, as indicated from the positive objective values.

"Unsatisfied" optimal results at $k_0=(k_0^{opt}-1)$ for some instances of $c_\varpi, c_\delta, i$ values.
Instance $k_0^{*}$ $~k_0~$ Parameters Error Terms Objective
$c_{\varpi}$ $~c_{\delta}~$ $~i~$ $\varpi$ $~\delta~$ $~\delta'~$ $~A~$ $~\kappa_1~$ $~\kappa_2~$ $~\kappa_3~$
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 $k_0=(k_0^{opt}-1)$ for some instances of $c_\varpi, c_\delta, i$ values, with an additional condition that $\left(1- \frac{j_{k_0-2}^2}{k_0(k_0-1)(1 + 4 \varpi)}\right) \ge 0$. Note that these $k_0$ values are more unsatisfied, as indicated from the larger positive objective values that that in the previous table.

More conservative "unsatisfied" results at $k_0=(k_0^{opt}-1)$ for some instances of $c_\varpi, c_\delta, i$ values.
Instance $k_0^{*}$ $~k_0~$ Parameters Error Terms Objective
$c_{\varpi}$ $~c_{\delta}~$ $~i~$ $\varpi$ $~\delta~$ $~\delta'~$ $~A~$ $~\kappa_1~$ $~\kappa_2~$ $~\kappa_3~$
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 $~c_\varpi$, a theoretical lower bound of $~k_0$, called $k_0^*$, can be obtained by assuming that all error terms $~\kappa_1$, $~\kappa_2$, and $~\kappa_3$ could be completely ignored. This table gives the computational results of $k_0^*$ for $c_\varpi \lt 87$.

$~c_\varpi~$ 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