Primality test and primes enumeration using odd numbers indexation

TL;DR

This paper introduces a novel indexing method for odd numbers to improve primality testing and prime enumeration, enhancing computational efficiency in classical algorithms like Atkin and Pritchard sieves.

Contribution

The paper presents a new odd number indexing scheme and adapts existing primality algorithms to leverage this for faster computation.

Findings

Reduced input size leads to constant-time improvements

Applied indexing to Atkin sieve with similar efficiency gains

Extended technique to Pritchard wheel sieve with comparable results

Abstract

Odd numbers can be indexed by the map k(n)=(n-3)/2, n belonging to 2N+3. We first propose a basic primality test using this index function that was first introduced in article (8). Input size of operations is reduced which improves computational time by a constant. We then apply similar techniques to Atkin prime-numbers sieve which uses modulus operations and finally to Pritchard wheel sieve, in both case yielding similar results.