Hexagon

math.NT — Number Theory

Linnik's constant is at most $3.99$

Contributed by Eric Naslund

Let $P(a,q)$ denote the least prime congruent to $a$ modulo $q$ , where $(a,q)=1$ . We give a computer-assisted proof that $P(a,q)\ll q^{3.99}$ , improving the exponent $5$ of Xylouris. More precisely, $P(a,q)<q^{3.99}$ for every sufficiently large $q$ , uniformly in $a$ . The main new ingredient is 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. 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 $14$ and $21$ 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 $4453$ root cases and $4788$ terminal cases, each closed by its own linear program. On $4{,}196{,}879$ threshold boxes these programs give $29{,}397{,}336$ linear relaxations, and exact integer certificates for all of them bound the normalized zero sum strictly below~ $1$ . 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).

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hexagon:2610.00016v1

math.CO — Combinatorics

An Exponent of 1.04273 for the Unit Distance Problem

Contributed by Eric Naslund

Let $u(U)$ count the unordered pairs at distance one in a finite planar set $U$ . We construct finite sets $U_j$ with $|U_j|\to\infty$ and $u(U_j)/|U_j|^{1.04273}\to\infty$ . The largest exponent previously claimed, $1.0358$ , is in the author's unpublished manuscript. The method is the number-field construction of OpenAI and Sawin: unit distances come from elements of relative norm one in quadratic extensions, and the fields come from an infinite pro- $2$ class field tower. The new ingredients are quadratic extensions of mixed signature, with an exact average over their norm-one units, and a tower over the real quadratic field $\Q(\sqrt{241})$, in which $2$ , $3$ and $5$ split. Its Golod--Shafarevich function contains two copies of the local conditions at these primes but only one constant term, and the extra room lets the tower be ramified only above $2$ , $3$ and $5$ ; the root discriminant of its fields is about $286$ . The relative zeta value is bounded through the zeta function of the degree- $512$ field generated over $\Q(\sqrt{241})$ by the square roots of its $\{2,3,5\}$ -units, which is the product of the Dedekind zeta function of $\Q(\sqrt{241})$ and $255$ quadratic Hecke $L$ -functions. Finite facts and numerical inequalities are certified by exact computation and interval arithmetic. A significant portion of this work was verified in Lean, reducing the result with exponent $1.0427$ to an explicit zeta function inequality (Palomar registry, PALOMAR-2026-10-01-000018, version 1).

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Submitted works

Linnik's constant is at most 3.993.993.99

An Exponent of 1.04273 for the Unit Distance Problem

Linnik's constant is at most $3.99$