Eratosthenes sieve supports the $k$-tuple conjecture

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.