Eratosthenes sieve supports the $k$-tuple conjecture

TL;DR

This paper models Eratosthenes sieve as a dynamic system to support the k-tuple conjecture, showing that admissible constellations of gaps appear and persist, aligning with Hardy-Littlewood estimates, and introduces primorial coordinates for tracking them.

Contribution

It demonstrates that all admissible constellations occur in Eratosthenes sieve and introduces a new notation for tracking their locations, strengthening the connection to the k-tuple conjecture.

Findings

Admissible constellations appear and persist in the sieve.

Population of constellations aligns with Hardy-Littlewood estimates.

Introduces primorial coordinates for tracking constellations.

Abstract

Viewing Eratosthenes sieve as a discrete dynamic system, we show that every admissible instance of every admissible constellation of gaps arises and persists in Eratosthenes sieve. For an admissible constellation of length J, we show that its population across stages of the sieve is consistent with the Hardy and Littlewood estimates from 1923. This work strongly connects Eratosthenes sieve to the k-tuple conjecture, and it provides a compact notation, primorial coordinates, for tracking the locations of admissible instances for a constellation.