Primitive roots for every admissible integer base

What OpenAI claims

Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least c_a x/(log x)² primes in every sufficiently large interval (x,2x) have primitive root a, with c_a > 0.

Papers (2)

1. Primitive roots for every admissible integer base

   4 October 2026 · Preprint, not peer reviewed · PDF

   We prove the infinitude assertion in Artin's primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least c_a x/(log x)² primes in (x,2x) with primitive root a, for some c_a > 0 and every sufficiently large x.

2. Simultaneous primitive roots: a conditional lower bound for prime bases

   4 October 2026 · Preprint, not peer reviewed · PDF

   For every fixed finite set of distinct positive primes, we prove that at least cx/(log x)² primes in (x,2x) have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.