OpenAI · 029-01 · 2026-10-04 · Number theory

Primitive roots for every admissible integer base

preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026

Family 029. Artin's primitive root conjecture: infinitude for every base.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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 $c_{a} x / (lo g x)^{2}$ primes in every sufficiently large interval $(x, 2 x)$ have primitive root $a$ , with $c_{a} > 0$ .

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 $c_{a} x / (lo g x)^{2}$ primes in $(x, 2 x)$ with primitive root a, for some $c_{a} > 0$ and every sufficiently large x.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.