Squarefree smooth numbers and Euclidean prime generators

TL;DR

This paper demonstrates that for primes greater than 7, all residues can be represented by squarefree numbers with bounded prime factors, and explores applications to recursive prime generation methods inspired by Euclid.

Contribution

It introduces a novel representation of residues using squarefree numbers with prime factor constraints and applies this to recursive prime generator constructions.

Findings

Every residue mod p (p > 7) can be represented by a squarefree number with prime factors ≤ p

Provides new methods for recursive prime generation based on these representations

Establishes connections between squarefree numbers and Euclidean prime generation techniques

Abstract

We show that for each prime p > 7, every residue mod p can be represented by a squarefree number with largest prime factor at most p. We give two applications to recursive prime generators akin to the one Euclid used to prove the infinitude of primes.