Evaluating a Deterministic Alternative to Probabilistic Coprime Set Generation

Abstract: In distributed systems and cryptography, generating mutually coprime sets typically relies on probabilistic rejection sampling, an approach bounded by unpredictable loops. This paper explores an O(1)O(1) O ( 1 ) deterministic alternative utilizing p-adic valuations and primorial offsets, transforming coprime generation from a trial-and-error search into a pure arithmetic geometry solution, reducing algorithmic complexity to a strict, single-pass operation. 1. The Euclidean Algorithm Rabbit Hole As software engineers, we rely heavily on the predictability of our algorithms. We prefer strictly bounded execution times and deterministic behavior. However, while brushing up on the classic Euclidean algorithm for finding the Greatest Common Divisor (GCD), I fell down a number theory rabbit hole that highlighted a surprising inefficiency in how we handle coprime integers in computer science. Two integers are coprime if their greatest common divisor is strictly 1 ( gcd⁡(a,b)=1\gcd(a,