Combinatorics of the gaps between primes

TL;DR

This paper develops a recursive model for the gaps between primes using Eratosthenes sieve, providing evidence for longstanding conjectures and revealing new insights into prime distributions and constellations.

Contribution

It introduces a recursive framework modeling prime gaps and constellations within Eratosthenes sieve, supporting conjectures like Polignac's and Hardy-Littlewood's.

Findings

Every even gap occurs in the sieve, supporting Polignac's conjecture.

Asymptotic populations align with Hardy-Littlewood's Conjecture B.

All feasible constellations of equal gaps occur, including those in arithmetic progression.

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. The populations of gaps and constellations across stages of Eratosthenes sieve are modeled exactly by discrete dynamic systems. These models and their asymptotic behaviors provide evidence on a number of open problems regarding gaps between prime numbers. For Eratosthenes sieve we show that the analogue of Polignac's conjecture is true: every gap $g=2k$ does occur in the sieve, and its asymptotic population supports the estimates made in Hardy and Littlewood's Conjecture B. A stronger form of Polignac's conjecture also holds for the sieve: for any gap $g=2k$, every feasible constellation $g,g,\ldots,g$ occurs; these constellations correspond to consecutive primes in arithmetic progression. The models also provide evidence toward resolving a series of questions posed by Erd\"os and Tur\'an.