Cycles and Patterns in the Sieve of Eratosthenes

TL;DR

This paper uncovers recurring patterns, symmetries, and cycles in the Sieve of Eratosthenes, extending these insights to general coprime sets and Euler's phi-function, revealing underlying distributional structures.

Contribution

It introduces a novel analysis of patterns and cycles in the sieve process and generalizes these to arbitrary prime sets and Euler's phi-function, offering new distributional insights.

Findings

Identification of symmetries and cycles in the sieve process

Generalization to numbers coprime to arbitrary prime sets

Insights into the distribution of Euler's phi-function values

Abstract

We describe recurring patterns of numbers that survive each wave of the Sieve of Eratosthenes, including symmetries, uniform subdivisions, and quantifiable, predictive cycles that characterize their distribution across the number line. We generalize these results to numbers that are relatively prime to arbitrary sets of prime numbers and derive additional insights about the distribution of integers counted by Euler's phi-function.