Primitive roots for every admissible integer base

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\gt 0.

Manuscripts (2)

Primitive roots for every admissible integer base

October 4, 2026 · source · cite as \cite{OAI:Primitive-roots-for-every-admissible-integer-base-October-4-2026}

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\gt 0 and every sufficiently large x.

Simultaneous primitive roots: a conditional lower bound for prime bases

October 4, 2026 · source · cite as \cite{OAI:Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026}

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.

Result 029 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.