HEXAGONbeta

Linnik's constant is at most 3.993.99

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Eric Naslund ↗

SubmitterEric Naslund

Version 1 / Oct 02, 2026 / CC BY 4.0

Abstract

Let P(a,q)P(a,q) denote the least prime congruent to aa modulo qq, where (a,q)=1(a,q)=1. We give a computer-assisted proof that P(a,q)≪q3.99P(a,q)\ll q^{3.99}, improving the exponent 55 of Xylouris. More precisely, P(a,q)<q3.99P(a,q)<q^{3.99} for every sufficiently large qq, uniformly in aa. The main new ingredient is a graded near-density estimate for zeros of Dirichlet LL-functions, a weighted form of Heath-Brown's Lemma~12.1 of the kind he asked for in 1992. We combine it with the zero-location estimates and far-density method of Heath-Brown and Xylouris. The second novelty is the scale of the case analysis. Heath-Brown and Xylouris closed their final case analyses with 1414 and 2121 main cases, each by a chain of inequalities evaluated in floating point. In this paper, with the help of advanced AI models, we can push this much further. The middle range of the first zero is divided into 44534453 root cases and 47884788 terminal cases, each closed by its own linear program. On 4,196,8794{,}196{,}879 threshold boxes these programs give 29,397,33629{,}397{,}336 linear relaxations, and exact integer certificates for all of them bound the normalized zero sum strictly below~11. Exceptional zeros and the remaining exterior range are treated separately. This case analysis is a kind of systematic brute force enabled by AI: an analysis of this size, with parameters tuned to each case, would be very laborious or nearly impossible to carry out by hand, and it lets the new estimate be applied separately in each case. The near-density lemma, certificate soundness, and a conditional passage from a certified case to a prime are formalized in Lean, assuming published analytic inputs and specified facts about zeros (PALOMAR-2026-10-01-000020 v1).

Provenance statement

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. The certificates are checked by three separate programs, one of them formally verified, and states precisely which parts of the proof are formally verified in Lean and which have conventional proofs only.
FormalizationsPalomar

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
ChatGPTVersion Astra 6
XAi
GrokVersion 4.7

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  1. v1Initial depositCurrentOct 02, 2026