On Polignac's Conjecture

TL;DR

This paper proves that every even number gap occurs infinitely often in Eratosthenes sieve and that their relative frequencies align with Hardy-Littlewood's conjectures for prime gaps.

Contribution

It generalizes a recursion for gaps in Eratosthenes sieve to show all even gaps occur infinitely often and match predicted ratios.

Findings

Every even gap occurs infinitely often in Eratosthenes sieve.

The ratio of gaps g=2n to g=2 converges to Hardy-Littlewood estimates.

Results support conjectures about prime gaps using sieve-based methods.

Abstract

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known as constellations. As the recursion proceeds, adjacent gaps within longer constellations are added together to produce shorter constellations of the same sum. These additions or closures correspond to removing composite numbers that are divisible by the prime for that stage of Eratosthenes sieve. Although we don't know where in the cycle of gaps a closure will occur, we can enumerate exactly how many copies of various constellations will survive each stage. In this paper, we broaden our study of these systems of constellations of a fixed sum. By generalizing our methods, we are able to demonstrate that for every even number $2n$ the gap $g=2n$ occurs infinitely often through the stages of Eratosthenes sieve. Moreover, we show that asymptotically the ratio of the number of gaps $g=2n$ to the number of gaps $g=2$ at each stage of Eratosthenes sieve converges to the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper.