All Prime Numbers Have Primitive Roots

TL;DR

This paper provides a constructive proof demonstrating that every prime number has at least one primitive root, building on prior ACL2 community work to formalize this fundamental number theory fact.

Contribution

It offers a formal, constructive proof that all primes possess primitive roots, extending previous work within the ACL2 formal verification community.

Findings

Confirmed that every prime has a primitive root

Formalized the proof within ACL2 system

Extended prior ACL2 work on primitive roots

Abstract

If p is a prime, then the numbers 1, 2, ..., p-1 form a group under multiplication modulo p. A number g that generates this group is called a primitive root of p; i.e., g is such that every number between 1 and p-1 can be written as a power of g modulo p. Building on prior work in the ACL2 community, this paper describes a constructive proof that every prime number has a primitive root.