Odlyzko–Poonen conjecture (1993) proved unconditionally: 'The proofs are due to GPT-6 Astra', which also wrote a 22,000-line Lean formalisation
Constantin Kogler posted an unconditional proof of the Odlyzko–Poonen conjecture: a random monic 0/1 polynomial with constant term 1 is irreducible over Q with probability tending to 1 (arXiv 2609.26771, 22 Sep 2026). Breuillard and Varjú had proved this in 2019 only under the Generalized Riemann Hypothesis. The paper says the proofs are due to GPT-6 Astra. The model also formalised everything in Lean (about 22,000 lines in about 30 hours), and Emmanuel Breuillard checked the proof.
Key facts
- Conjecture: Odlyzko and Poonen (1993); Konyagin (1999) gave a c/log n lower bound; Breuillard–Varjú (2019) proved it under GRH; Bary-Soroker–Koukoulopoulos–Kozma (2023) handled coefficients from at least 35 consecutive integers
- New: P(P_n irreducible) → 1 unconditionally, with P(reducible) = sqrt(2/(πn)) + O(1/n)
- Machine Contribution Statement: 'The proofs are due to GPT-6 Astra. The author rewrote and checked the arguments'; first found on 14 Sep 2026, after the model had proved a related result on Bernoulli convolutions (first under GRH, then unconditionally)
- 'GPT-6 Astra formalized all results from this paper and all necessary results from previous work. The formalization took around 30 hours and around 22'000 lines of code were written' (repository ckkogler/odlyzko-poonen-lean)
- Kogler thanks Emmanuel Breuillard 'for checking the proof', and thanks Breuillard and Péter Varjú for help with the writing
Science result
- Field
- mathematics / number theory / probability
- Problem
- Odlyzko–Poonen conjecture on irreducibility of random 0/1 polynomials (open since 1993)
- Result
- Unconditional proof that random 0/1 polynomials are irreducible with probability tending to one, with the sharp asymptotic for the reducible probability.
- AI system
- GPT-6 Astra
- Human role
- AI-generated proofs and formalisation; the author directed the work, rewrote and checked it; an expert (Breuillard) checked the proof
- Verification
- Formal proof in Lean 4, plus expert check; preprint, not peer-reviewed
- Status
- confirmed
What happened
While studying Bernoulli convolutions with GPT-6 Astra, Kogler saw the model remove a GRH assumption from a related result. He then asked it to try the Odlyzko–Poonen conjecture. The model produced a proof and formalised it together with all the prior results it needed.
Why it matters
The unconditional case had stayed open after Breuillard and Varjú's conditional proof in 2019. The paper assigns the mathematics fully to the model and includes both a machine-checked proof and a check by a leading expert, which is unusually complete verification for an AI result.
Changelog
- 2026-09-30: created
Related events
- OpenAI releases GPT-6 Astra, its first GPT-6 model ★★★★★
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
Sources (2)
- paperarXiv 2609.26771: The Odlyzko–Poonen Conjecture on Irreducibility of Random Polynomials (Kogler)
- codeGitHub: ckkogler/odlyzko-poonen-lean
id: 2026-09-22-odlyzko-poonen-conjecture-proved · updated 2026-09-30 · open in the interactive timeline