© 2011 *by Alexei Kourbatov, JavaScripter.net/math*

Main article: *Maximal gaps between prime k-tuples*

The
Hardy-Littlewood *k*-tuple conjecture
allows one to predict the *average frequencies* of prime *k*-tuples near *p*,
as well as the approximate total counts of prime *k*-tuples below *x*.
Specifically, the Hardy-Littlewood *k-tuple constants* *H _{k}*

Frequency of k-tuples ≈H/ log_{k}^{k}por, equivalently,Total number of k-tuples belowx

Reciprocal Hardy-Littlewood constants multiplied by log^{k}*p* determine the average gap between prime *k*-tuples near *p*:

Expected average gap a=Clog_{k}^{k}p,where C= 1/_{k}H._{k}

Using the expected average gap *a*, we can estimate
maximal gaps *g _{k}* between prime

Expected maximal gap = a(log(p/a) −b),with .a=Clog_{k}^{k}p,b≈ 2/k

**Table.** Hardy-Littlewood constants for prime *k*-tuples up to *k* = 8.

Name of prime k-tuple |
Width | Pattern | Hardy-Littlewood constant H_{k} |
ReciprocalC = 1/_{k}H_{k} |
---|---|---|---|---|

2-tuple, twin primes, twins | 2 | 0 2 | 1.32032 | 0.757392 |

3-tuple, triplet (type A) | 6 | 0 4 6 | 2.85825 | 0.349864 |

3-tuple, triplet (type B) | 6 | 0 2 6 | 2.85825 | 0.349864 |

4-tuple, quadruplet | 8 | 0 2 6 8 | 4.15118 | 0.240895 |

5-tuple, quintuplet (type A) | 12 | 0 4 6 10 12 | 10.13179 | 0.09869924 |

5-tuple, quintuplet (type B) | 12 | 0 2 6 8 12 | 10.13179 | 0.09869924 |

6-tuple, sextuplet | 16 | 0 4 6 10 12 16 | 17.29861 | 0.05780811 |

7-tuple, septuplet (type A) | 20 | 0 2 6 8 12 18 20 | 53.97195 | 0.01852814 |

7-tuple, septuplet (type B) | 20 | 0 2 8 12 14 18 20 | 53.97195 | 0.01852814 |

8-tuple, octuplet (type A) | 26 | 0 2 6 8 12 18 20 26 | 178.26195 | 0.005609722 |

8-tuple, octuplet (type B) | 26 | 0 6 8 14 18 20 24 26 | 178.26195 | 0.005609722 |

8-tuple, octuplet (type C) | 26 | 0 2 6 12 14 20 24 26 | 475.36521 | 0.002103646 |

Hardy-Littlewood constants can be computed with a high precision;
see Tony Forbes *k*-tuplets pages for more information on this.

Copyright © 2011, Alexei Kourbatov, JavaScripter.net.