Constellations of gaps in Eratosthenes sieve

TL;DR

This paper extends a discrete model of Eratosthenes sieve to analyze constellations of gaps, revealing that certain arithmetic progressions of candidate primes occur infinitely often, supporting conjectures about prime distributions.

Contribution

It introduces a new discrete linear system modeling populations of constellations of gaps in Eratosthenes sieve, extending previous models to include arithmetic progressions.

Findings

Constellations of gaps follow predictable populations across sieve stages.

Every even gap with specific divisibility properties appears as a constellation.

Arithmetic progressions of candidate primes are shown to occur infinitely often in the sieve.

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. Over the last year we identified a discrete linear system that exactly models the population of any gap across all stages of the sieve. In August 2014 we summarized our results from analyzing this discrete model on populations of single gaps. This paper extends the discrete system to model the populations of constellations of gaps. The most remarkable result is a strong Polignac result on arithmetic progressions. We had previously established that the equivalent of Polignac's conjecture holds for Eratosthenes sieve -- that every even number arises as a gap in the sieve, and its population converges toward the ratio implied by Hardy and Littlewood's Conjecture B. Extending that work to constellations, we here establish that for any even gap $g$, if $p$ is the maximum prime such that $p\# \; | g$ and $P$ is the next prime larger than $p$, then for every $2 \le j_1 < P-1$, the constellation $g,g,\ldots,g$ of length $j_1$ arises in Eratosthenes sieve. This constellation corresponds to an arithmetic progression of $j_1+1$ consecutive candidate primes.