# Finding optimal k0 values

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== Code and data == | == Code and data == | ||

- | * [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding optimal <math> k_0</math> values | + | * [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding optimal <math> k_0</math> values]: Java, bash, and maple are needed. |

## Current revision

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

- () is a quantity such that every admissible -tuple has infinitely many translates which each contain at least two primes. It is then used for finding narrow admissible tuples.

- holds for some combinations of , and values, where means -tuply densely divisible, and are constants in the constraint on and , such that .

## Contents |

## Optimization Model

For a given set of , the basic optimization model is defined as:

and 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 values, respectively without and with using Deligne's theorem.

Instance | Parameters | Error Terms | Comment | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

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 for some instances of values.

Instance | Parameters | Error Terms | Objective | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

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 for some instances of values. Note that the results of these *k*_{0} values do not satisfy all constraints, as indicated from the positive objective values.

Instance | Parameters | Error Terms | Objective | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

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 for some instances of values, with an additional condition that . Note that these *k*_{0} values are more unsatisfied, as indicated from the larger positive objective values that that in the previous table.

Instance | Parameters | Error Terms | Objective | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

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 , a theoretical lower bound of , called , can be obtained by assuming that all error terms , , and could be completely ignored. This table gives the computational results of for .

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
*k*_{0}values: Java, bash, and maple are needed.