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A prime gap is the difference between two successive prime numbers. Prime gaps are casually thought to occur randomly. However, the “ k -tuple conjecture” suggests that prime gaps are non-random by estimating how often pairs, triples and larger groupings of primes will appear. The k -tuple conjecture is yet to be proven, but a very recent work presents a result that contributes to a confirmation of the k -tuple conjecture by finding unexpected biases in the distribution of consecutive primes. Here, we present another contribution to confirmation of the k -tuple conjecture based on statistical physics. The pattern we find comes in the form of a power law in the distribution of prime gaps. We find that prime gaps are proportional to the inverse of the chance of a number to be prime.

Prime numbers are divisible only by themselves and 1. Primes are the building blocks of the entire number line because all the other numbers are created by multiplying primes together. Thus, primes are the core of arithmetic.

Whether a number is prime or not is pre-determined, as evidenced by innumerous laws already proven. For instance, the prime number theorem states that the average length of the gap between a prime

Apart from 2 and 5, all prime numbers end in 1, 3, 7 or 9, and each of the four endings is supposedly equally likely. But the authors in Ref. [

The authors then show that the last-digit pattern can be explained by the groupings given by the k-tuple conjecture. However, as the primes tend to infinity, the pattern vanishes and the primes become genuinely random. Here, we contribute to the literature by presenting further evidence that the k-tuple conjecture can be true. In line with the authors in Ref. [

There is already substantial literature on primes adopting the statistical physics perspective. In line with our finding, the histograms in the distribution of gaps between primes divided into “congruence families” are shown to be scale invariant [

As the authors in Ref. [

Then we let

Number n | Prime? 0 = no, 1 = yes |
---|---|

1 | 0 |

3 | 1 |

7 | 1 |

9 | 0 |

11 | 1 |

13 | 1 |

17 | 1 |

19 | 1 |

21 | 0 |

23 | 1 |

27 | 0 |

29 | 1 |

31 | 1 |

33 | 0 |

37 | 1 |

… | … |

consider a Bernoulli random variable

To overcome this difficulty, we devised the following: Assume

Thus, for

This power law can be translated in terms of prime gaps, as in

Then, let

Prime number | Prime gap |
---|---|

2 | 2 |

3 | 1 |

5 | 2 |

7 | 2 |

11 | 4 |

13 | 2 |

17 | 4 |

19 | 2 |

23 | 4 |

29 | 6 |

... | ... |

successive Bernoulli trials are independent. Thus, the expected number of trials until the occurrence of a subsequent prime is given by:

However, because our Bernoulli process is nonstationary and the Bernoulli trials are not independent, one cannot expect an analytic solution such as that provided by Equation (3) to hold true. But one can empirically determine an analogous substitute for Equation (3) by considering estimates of

Using the nonlinear Kalman filter for this geometric process,

(considering a Bernoulli process). As can be seen,

Therefore, prime gaps are proportional to the inverse of the chance of a number to be prime.

Alternatively, consideration of the power law in Equation (2) yields:

Although the difference between two successive prime numbers is casually considered random, the k-tuple conjecture casts doubt on that. The k-tuple conjecture is yet to be

proven, but finding unexpected biases in the distribution of consecutive primes provides confirmation of the k-tuple conjecture. Motivated by a recent mathematical study [

Matsushita, R. and Da Silva, S. (2016) A Power Law Governing Prime Gaps. Open Access Library Journal, 3: e2989. http://dx.doi.org/10.4236/oalib.1102989