{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-artin-primitive-root-conjecture-infinitude/","md":"https://postcutoff.com/e/2026-10-06-openai-artin-primitive-root-conjecture-infinitude/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-06-openai-artin-primitive-root-conjecture-infinitude","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Artin's primitive root conjecture (infinitude)","deck":null,"takeaway":"Family 029 of OpenAI's 6 Oct 2026 release claims that every integer other than −1 and the squares is a primitive root modulo infinitely many primes, at least c·x/(log x)² of them in each interval (x, 2x). Before, this was known only under the generalized Riemann hypothesis (Hooley, 1967) or for all but at most two prime bases (Heath-Brown, so one of 2, 3, 5 works), not for any single named base such as 2. The argument rests on the release's own zero-free half-planes. Frank Calegari calls the argument interesting but 'not at all surprising' once quasi-RH is known. Manjul Bhargava has posted a positive-density strengthening.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Primitive roots for every admissible integer base (4 Oct)","url":"https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI: Simultaneous primitive roots, a conditional lower bound for prime bases","url":"https://github.com/openai/math/blob/main/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"Hexagon 2610.00228: Every nonsquare integer not equal to −1 is a primitive root modulo a positive lower density of primes (contributed by Manjul Bhargava)","url":"https://hexagonmath.org/abs/2610.00228v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":4,"title":"Hexagon 2610.00200: Cureton and Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions","url":"https://hexagonmath.org/abs/2610.00200v2","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":5,"title":"Scientific American: The most exciting claims from OpenAI's heap of new proofs","url":"https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/","type":"press","group":"press","domain":"scientificamerican.com"},{"n":6,"title":"Persiflage (Frank Calegari): A Brief Digest of Some New OpenAI Results (8 Oct, item 4 and comments)","url":"https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","type":"discussion","group":"reactions","domain":"galoisrepresentations.org"},{"n":7,"title":"oai-math-revised: community notes on the primitive-root manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release claims the infinitude part of Artin's primitive root conjecture (1927) for every admissible base, including 2; Bhargava and others build on it","summary":"\"Primitive roots for every admissible integer base\" (OpenAI, 4 Oct 2026, family 029) proves, for every integer a that is neither −1 nor a square, that at least c_a·x/(log x)² primes in (x, 2x) have a as a primitive root. This gives the infinitude assertion of Artin's 1927 conjecture for every admissible base, including 2, though not the conjectured density. The proof combines a zero-free half-plane for Hecke L-functions, of the kind behind the release's quasi-Riemann hypothesis, with sieve estimates for primes p whose p − 1 has a controlled shape. It is unrefereed and not formalized. Follow-ups by Cureton–Thorner and by Manjul Bhargava (Hexagon, partly AI-written) extend the zero-free region and upgrade the count to a positive lower density.","key_facts":["Theorem 1.1 (verbatim): 'For every admissible integer a, there are constants ca > 0 and xa ≥ 2 such that, for every real x ≥ xa, #{p prime : x < p < 2x, p ∤ a, ordp(a) = p − 1} ≥ ca x/(log x)².' The paper: 'The predicted counting scale is x/ log x … In particular, it proves the infinitude assertion for each fixed admissible integer, including the prescribed base 2.' 92 pages, dated 4 Oct 2026; companion 'Simultaneous primitive roots: a conditional lower bound for prime bases' (21 pp.)","Prior state (paper, §1.1): Hooley proved Artin's predicted asymptotic assuming the Riemann hypothesis for the relevant Dedekind zeta functions (1967); unconditionally, Gupta and Murty gave a set of thirteen integers one of which works, and Heath-Brown 'proved that at most two prime bases fail the infinitude assertion, so at least one of 2, 3, 5 satisfies it' (verbatim), without identifying which","Method: a zero-free half-plane for Hecke L-functions over cyclotomic fields from a cubic theta input, a quadratic large sieve, and 'marked Type II estimates' to count primes with p − 1 equal to a small cofactor times one large prime. A blog commenter ('DH', 9 Oct) noted that the write-up contains a zero-free region Re s > 999999/1000000 for finite-order Hecke characters of cyclotomic fields containing ζ12. No Lean scope note","Frank Calegari (Persiflage, 8 Oct, verbatim): 'the argument is a little bit different, but also not at all surprising'; 'this is an interesting type of argument which I don’t find particularly surprising. But no one was going to write this paper, which depends on the quasi-Riemann hypothesis for Dirichlet L-functions, without knowing anything about general Dedekind L-functions.'","Follow-ups: Dean Cureton and Jesse Thorner (Hexagon, 12 Oct) prove a quasi-Riemann hypothesis for every Dedekind zeta function (Re s > 31/32), building on the zero-free argument of this family (see 2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover). Manjul Bhargava (Hexagon 2610.00228, 14 Oct, mixed human and AI text): 'a positive lower density of primes have a as a primitive root', using Cureton–Thorner's half-plane and 'the marked Type II estimates of OpenAI'; 'The argument gives a lower bound rather than the exact Artin density.' His provenance note mentions handwritten notes, 'a conversation with Peter Sarnak', GPT-6 and a Lean check of the deductions only","Scientific American (8 Oct) listed Artin's conjecture among the release's other important number-theory results, without expert comment"],"key_numbers":[],"tags":["math","number-theory","analytic-number-theory","primitive-roots","artin-conjecture","l-functions","openai","openai-math-release","unverified","claim"],"science":{"field":"mathematics","subfield":"analytic number theory","problem":"Artin's primitive root conjecture (1927): every integer other than −1 and the squares is a primitive root modulo infinitely many primes","result":"Claimed: for every admissible a, at least c_a·x/(log x)² primes in (x, 2x) have primitive root a (infinitude, not the conjectured density); a positive lower density was added by a follow-up","open_since":"1927","ai_system":["OpenAI internal model (openai/math release)"],"human_role":"Autonomous AI output released by OpenAI; follow-ups by human mathematicians with AI assistance","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed manuscript, no Lean formalization; follow-ups take its estimates as inputs"},"status":"pending","shock":"No single base, not even 2, was known unconditionally to be a primitive root for infinitely many primes."},"body_md":"## What happened\n\nA number a is a primitive root modulo a prime p when its powers run through every nonzero residue. Emil Artin conjectured in 1927\nthat every integer other than −1 and perfect squares is a primitive root for infinitely many primes, with a predicted density.\nChristopher Hooley proved this in 1967 assuming the generalized Riemann hypothesis. Without that assumption, Gupta, Murty and\nHeath-Brown showed in the 1980s that at most two prime bases can fail, so one of 2, 3 and 5 works, but nobody could name a single\nbase, not even 2, that provably works.\n\nOpenAI's manuscript claims the infinitude statement for every admissible base, with a lower bound of order x/(log x)² primes in each\ndyadic range rather than the predicted x/log x. As Calegari reconstructs it, the idea is to find primes p for which p − 1 has a\nspecial shape, a small cofactor times a large prime, so that only a few cases of Hooley's argument need a zero-free region. That\nregion comes from the same automorphic machinery as the release's quasi-Riemann hypothesis.\n\n## Why it matters\n\nArtin's conjecture is one of the classical problems of number theory. A proof of its qualitative part for every base, without the\nRiemann hypothesis, would settle what has been open for almost a century, although the exact Artin density remains conjectural.\n\n## Reactions and follow-ups\n\nCalegari found the argument natural once quasi-RH is available, but noted that nobody would have written it without that input. Dean\nCureton and Jesse Thorner extended the zero-free half-plane to every Dedekind zeta function within a week, and Manjul Bhargava posted a\nnote (written with GPT-6, deductions checked in Lean) upgrading the count to a positive proportion of primes. No independent check of\nthe 92-page proof itself has been published.","disputed":[{"label":"Verification","text":"Unrefereed manuscript, no Lean formalization; follow-ups take its estimates as inputs","sources":[]}],"related":[{"id":"2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover","url":"https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/","date":"2026-10-12","date_precision":"day","short_title":"Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)","deck":null,"takeaway":"It is one of the first results to build directly on OpenAI's claimed quasi-Riemann breakthrough.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Axiom Math"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-quasi-riemann-hypothesis-openai","url":"https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the quasi-Riemann hypothesis","deck":"No zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)","takeaway":"If correct, this is the largest advance on the zeros of ζ since the 19th century, and it would remove the Siegel-zero obstruction that runs through modern analytic number theory.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":19,"official":14,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (families 001–125; manuscript introduction, Persiflage digest and comments, Bhargava's and Cureton–Thorner's Hexagon pages read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Families 001–125; manuscript introduction, Persiflage digest and comments, Bhargava's and Cureton–Thorner's Hexagon pages read","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-artin-primitive-root"}