Primitive roots for every admissible integer base
Primitive roots for every admissible integer base is the subject of a manuscript dated 4 October 2026 in OpenAI's public mathematics collection (github.com/openai/math). Artin's primitive root conjecture predicts that every integer other than −1 that is not a perfect square is a primitive root modulo infinitely many primes, and indeed modulo a positive proportion of primes. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original result
- Claim. For every integer a that is neither −1 nor a square, there is ca > 0 such that, for all large x, the interval between x and 2x contains at least cax/(log x)2 primes with primitive root a; the manuscript states that this proves the infinitude assertion of Artin's conjecture. Source: the manuscript Primitive roots for every admissible integer base, dated 4 October 2026.
- Formalization. The collection's index (commit
adc7f12) lists no Lean formalization for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 14 October 2026) |
|---|---|---|---|---|
2026-10-09 (latest commit 0c827db, 14 October 2026) |
Yao Xu, cadamcat/artin-primitive-roots-lean4 (GitHub) | Formalization (Lean) | A Lean 4 proof that for every integer a ≠ −1 that is not a perfect square, the set of primes p for which a has multiplicative order p − 1 modulo p is infinite, following the manuscript. The README states that the proofs are verified by the Lean 4 kernel and records a Comparator check with two kernels. | Author-reported; the README states that the work was developed with AI assistance. |
| 2026-10-12 | D. Cureton, J. Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions, Hexagon 2610.00200v1 | Related extension | For every number field K, the Dedekind zeta function of K has no zeros with real part greater than 1 − 1/6000. The abstract describes the method as following the strategy of recent work on Artin's primitive root conjecture. | Posted on Hexagon (v1); the page states that the proofs and the Lean formalization were generated by AxiomProver. |
| 2026-10-14 | M. Bhargava, Every nonsquare integer not equal to −1 is a primitive root modulo a positive lower density of primes, Hexagon 2610.00228v1 | Extension | For every integer a that is neither a square nor −1, a positive lower density of primes have a as a primitive root. A zero-free half-plane Re s > 1 − 1/N for all Dedekind zeta functions gives a lower proportion of at least ca/(N log N); the Cureton–Thorner theorem supplies N = 6000, so the conclusion is unconditional. The abstract states that the proof uses the marked Type II estimates of the OpenAI manuscript and gives a lower bound rather than the exact Artin density. | Posted on Hexagon (v1), labelled as a mix of human-written and AI-generated text with human understanding of some parts. The provenance statement says the deductions were formalized in Lean with the help of GPT-6, with the cited theorems of Cureton–Thorner and OpenAI as explicit hypotheses. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.