Patterns among the Primes: A study of Eratosthenes sieve Paperback – June 4, 2022

The authors study the gaps between primes by studying the cycles of gaps associated with Eratosthenes Sieve. Viewing the sieve as a discrete dynamic system, the authors develop exact population models for gaps and constellations across stages of Eratosthenes sieve. This approach through discrete mathematics provides a constructive complement for studying these topics, and they inspire intuitions about several open conjectures.The twin-prime conjecture & Polignac's conjecture. Every even number arises as a gap in Eratosthenes sieve, and its asymptotic population, relative to other gaps, depends only on its prime factors. k-tuple conjecture. Every admissible constellation arises and persists across the stages of Eratosthenes sieve.Consecutive primes in arithmetic progression. Every admissible repetition of a gap arises in the sieve, and the asymptotic population of this repetition, relative to other constellations of the same length, depends only on the prime factors of the gap and the length of the repetition.Convexity conjecture. The study provides coordinates within the sieve for some of the counterexamples identified by Engelsma.This investigation includes a statistical study of how well the gaps between primes reflect the relative populations of gaps in Eratosthenes sieve. Read more