Cycles and Patterns in the Sieve of Eratosthenes, Potential Twin Primes

TL;DR

This paper investigates the distribution and patterns of potential twin primes, which are pairs of integers with specific coprimality properties, revealing cyclical patterns and providing a counting formula to understand their distribution.

Contribution

It introduces a novel analysis of potential twin primes, detailing their cyclical patterns and deriving a formula for their enumeration, advancing understanding of prime pair distributions.

Findings

Identification of cyclical patterns in potential twin primes

Derivation of a formula for counting potential twin primes

Insights into the distribution of twin primes across the number line

Abstract

This paper analyzes the emergence and distribution of potential twin primes, pairs of integers that are both relatively prime to the first n primes or to a given set M of primes, and which are the breeding grounds of true twin primes. It describes cyclical patterns of their location across the number line and provides a formula for counting them.