Linnik’s constant ≤ 3.99, written by Claude Opus 5.5 and GPT Astra
Awaiting review
Importance: major (4 of 5)The takeaway
Linnik proved in 1944 that the least prime congruent to a mod q is at most q^L for an absolute constant L. Heath-Brown brought L down to 5.5 in 1992 and Xylouris to 5. On 2 Oct Eric Naslund posted on Hexagon a computer-assisted proof that L ≤ 3.99. It rests on a weighted form of a Heath-Brown lemma that Heath-Brown said no neat way of achieving ‘seems available’. The paper says it was written, and the research carried out, by Claude Opus 5.5 and GPT Astra 6 under Naslund’s prompting and supervision, with some formalization by Grok. Two components are Lean-verified on Palomar; the final bound itself is not. Hexagon lists human understanding as ‘some parts’, and we found no expert assessment as of 29 Oct.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 29 October 2026
Your AI and this story
- GPT-6 Astra155 days after its cutoff
- Claude Opus 5.594 days after its cutoff
- Gemini 4 Argon185 days after its cutoff
- Grok 4.7124 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 94 days before it.
Key facts
- Claim (abstract, verbatim): ‘We give a computer-assisted proof that P(a,q) ≪ q^3.99, improving the exponent 5 of Xylouris. More precisely, P(a,q) < q^3.99 for every sufficiently large q, uniformly in a.’
- New ingredient (abstract): ‘a graded near-density estimate for zeros of Dirichlet L-functions, a weighted form of Heath-Brown’s Lemma 12.1 of the kind he asked for in 1992.’ The Palomar record adds that Heath-Brown ‘remarks (section 16, item 9) no neat way of achieving’ it ‘seems available’
- Scale: Heath-Brown and Xylouris ‘closed their final case analyses with 14 and 21 main cases, each by a chain of inequalities evaluated in floating point’; the new proof uses a far larger certified case analysis (leaf linear programs with exact certificates)
- Provenance statement (verbatim): ‘This paper was written by Claude Opus 5.5 and GPT Astra 6 under my prompting and supervision. The research was carried out by the same models under my prompting and supervision, with some of the formalization work also done by xAI’s Grok. My recommendation is for the reader to explore this paper with AI as well, and use their favorite model to ask the paper questions.’ Hexagon labels: ‘Primarily AI-generated text’, ‘Human understanding: some parts’
- Formal status (Palomar abstract): two components are Lean-verified, the graded near lemma and a sound exact checker for the leaf-LP certificates; ‘That bound itself is NOT formally verified here: the paper’s zero-location arguments, case analysis and exterior regimes have conventional proofs and computer checks only.’ The certificates ‘are checked by three separate programs, one of them formally verified’
- Same contributor: ‘Square-difference-free sets of exponent 0.7580758’ (Hexagon 2610.00151, 8 Oct), D(N) ≥ N^0.7580758…, above Krachun’s 0.752796…; ‘written by GPT-6-Astra and revised by Claude Opus 5.5 under my prompting and supervision’; Naslund understands ‘the core argument’ but has ‘not carefully read some of the optimizations’
- Not checked: no referee, no expert comment found (web, Hexagon, HN) as of 29 Oct
What happened
Every reduced residue class mod q contains primes, and Linnik showed in 1944 that the first one is below q^L for a fixed L. Pushing L down is a benchmark of analytic number theory, because it depends on how well one controls zeros of Dirichlet L-functions near 1. Heath-Brown reached 5.5 in 1992 with an elaborate case analysis, and Xylouris lowered it to 5.
On 2 Oct Eric Naslund deposited on Hexagon a paper claiming 3.99. Its key lemma is a weighted density estimate that Heath-Brown had wished for, and its case analysis is closed by exact certificates checked by three separate programs. Naslund writes that Claude Opus 5.5 and GPT Astra 6 did the research and the writing under his prompting and supervision. A Palomar record holds Lean proofs of the new lemma and of the certificate checker’s soundness. It states that the bound itself is not formally verified.
Why it matters
If correct, it is the largest improvement to Linnik’s constant since Heath-Brown’s 1992 paper, and it comes from model-driven research with only partial human understanding claimed. The partial formalization shows where to look: the zero-location arguments and the case analysis outside the certified leaves still need a human or machine check.
What is disputed or not yet verified
| Verification | Hexagon preprint; computer-assisted certificates checked by three programs; two components Lean-verified (Palomar), the final bound not formalized |
|---|
Sources
3 sources from 2 sites. Numbers match the chips in the text.
3 sources: 3 primary
Primary
- Hexagon 2610.00018: Linnik’s constant is at most 3.99 (Naslund; Claude Opus 5.5, GPT Astra 6, Grok)hexagonmath.org, paper
- Palomar PALOMAR-2026-10-01-000020: Lean components of the Linnik 3.99 proofpalomar-registry.org, code
- Hexagon 2610.00151: Square-difference-free sets of exponent 0.7580758 (Naslund)hexagonmath.org, paper
Changes
- Filed (Hexagon records of 2610.00018 and 2610.00151, Palomar API record; found in a first full read of the Hexagon API)