Hexile Sieve Analysis of Prime and Composite Integers

TL;DR

This paper introduces the Hexile sieve, a novel method based on modulo 6 equivalence classes, to analyze primes and composites, reducing search space and providing new insights into their properties and distribution.

Contribution

The paper presents a new sieve method that narrows prime search to one-third of natural numbers and derives diophantine equations to distinguish primes from composites.

Findings

Reduces prime search space to one-third of natural numbers

Provides diophantine equations to differentiate primes and composites

Suggests a new route to prime counting functions

Abstract

Here we demonstrate a sieve for analysing primes and their composites, using equivalence classes based on the modulo 6 return value as applied to the Natural numbers. Five features of this 'Hexile' sieve are reviewed. The first aspect, is that it narrows the search for primes to one-third of the Natural numbers. The second feature is that we can obtain from the equivalence class formulae, a property of its diophantine equations to distinguish between primes and composites resulting from multiplication of these primes. Thirdly we can from these diophantine formulations ascribe a non-random occurence to not only the composites in the two equivalence classes but by default and as a consequence : non-randomness of occurence to the resident primes. Fourthly we develop a theoretical basis for sieving primes. Of final mention is that the diophantine equations allows another route to a prime counting function using combinatorics or numerical analysis.