Eratosthenes sieve and the gaps between primes

TL;DR

This paper introduces a recursion based on Eratosthenes sieve that explicitly enumerates prime gaps and constellations, providing evidence for longstanding conjectures like twin primes and Polignac's conjecture.

Contribution

It develops a new recursion for prime gaps within Eratosthenes sieve, offering insights into prime gap distributions and supporting several open conjectures.

Findings

Evidence supporting twin prime conjecture

Insights into prime gap distributions

Support for Polignac's conjecture

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 sequences are known as constellations. By studying this recursion on the cycles of gaps across stages of Eratosthenes sieve, we are able to provide evidence on a number of open problems regarding gaps between prime numbers. The basic counts of short constellations in the cycles of gaps provide evidence toward the twin prime conjecture and toward resolving a series of questions posed by Erdos and Turan. The dynamic system underlying the recursion provides evidence toward Polignac's conjecture and in support of the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper.