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Number Theory

math.NT — Number Theory

Works in math.NT

9 works

math.MG — Metric Geometry

A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

Contributed by Scott Kominers

We deduce a new asymptotic upper bound $K_{\mathrm{lat}}(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n$ on the maximal lattice kissing number $K_{\mathrm{lat}}(n)$ in dimension $n$ , using the explicit auxiliary functions constructed in OpenAI's "Ten Advances" preprint. The same bound holds for the average contact degree of a finite packing of congruent balls. The bound's base- $2$ exponential rate rounds to $0.3956$ , matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.

A mix of human-written and AI-generated textHuman understanding: all partsContact numbersDiscrete geometryEuclidean latticesFourier linear programming boundsGeometry of numbersKissing numbersSphere packing

Oct 02, 2026PDF ↗View work →

hexagon:2610.00017v1

math.NT — Number Theory

On the ternary pentagonal numbers conjecture

Contributed by Glenn Bruda

Communicated by Guy in 1994, the ternary pentagonal numbers conjecture of Blecksmith and Selfridge asserts that every integer larger than $33066$ is the sum of three positive pentagonal numbers. Prior to this note, it was even unknown whether every sufficiently large integer is the sum of three positive pentagonal numbers. We resolve this in the affirmative, using the landmark work of Duke and Schulze-Pillot on ternary quadratic forms to handle all sufficiently large integers $n$ with $v_3(24n+3)\leq8$ , and present an explicit lift to handle the $n$ with $v_3(24n+3)\geq9$ .

Primarily human-written textHuman understanding: all partsPentagonal numbersalmost universalityternary quadratic forms

Oct 02, 2026PDF ↗View work →

hexagon:2610.00018v1

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).

Primarily AI-generated textHuman understanding: some parts

Oct 02, 2026PDF ↗View work →

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).

Primarily AI-generated textHuman understanding: some parts

Oct 02, 2026PDF ↗View work →

hexagon:2609.00128v1

math.AG — Algebraic Geometry

On properties of Plancherel algebras

Contributed by Shurui Liu

We proved the predictions by Ben-Zvi, Sakellaridis, and Venkatesh that the Plancherel algebra $\mathrm{PL}_{X}$ is commutative and its loop-rotated version $\mathrm{PL}_{X,\hbar}$ is flat over $k[\hbar]$ .

Primarily AI-generated textHuman understanding: some partsPlancherel algebraRelative Langlandsgeometric Langlands

Sep 30, 2026PDF ↗View work →

hexagon:2609.00132v1

math.NT — Number Theory

Primitive Integer Solutions of x^4 + 3y^3 + 2z^3 = 0

Contributed by Avraham Eisenberg

We determine the primitive integer solutions of x^4 + 3y^3 + 2z^3 = 0. They are exactly (1,-1,1) and (-1,-1,1), where primitivity means gcd(x,y,z)=1. A fourth-power descent in Q(∛12) produces four plane quartics. A congruence modulo 4 excludes two of them for primitive input. We determine the rational points on the other two by elliptic descent, finite reduction sieves, and elliptic Chabauty. For the final quartic, an explicit nonzero Cassels–Tate pairing sharpens a rank-three descent bound to rank one. All finite certificates and their verification programs accompany the proof.

Primarily AI-generated textHuman understanding: no parts

Sep 29, 2026PDF ↗View work →

hexagon:2609.00130v1

math.NT — Number Theory

Primitive Integer Solutions of x^4 + 4y^3 + z^3 = 0

Contributed by Avraham Eisenberg

We determine all primitive integer solutions of the generalized Fermat equation x^4 + 4y^3 + z^3 = 0. Here primitive means gcd(x,y,z)=1. We prove that the only primitive solutions are (x,y,z) = (1,0,-1), (-1,0,-1). The proof begins with a fourth-power descent in the pure cubic field Q(∛2), reducing the problem to the rational points on two explicit smooth plane quartics of genus 3. These quartics admit maps to elliptic curves over Q(∛2). Exact 2-descent determines the relevant Mordell-Weil ranks and finite-index subgroups. A complete reduction sieve at 127 leaves four residue disks, and an elliptic Chabauty calculation proves that each disk contains exactly one rational point. Reconstruction of the original variables then leaves precisely the two stated primitive solutions. The finite computations are exact and reproducible from the certificates and verifier source included in the appendices, and the argument applies at all heights.

Primarily AI-generated textHuman understanding: no parts

Sep 29, 2026PDF ↗View work →

hexagon:2609.00119v1

math.NT — Number Theory

Hilbert-type inequality with primes

Contributed by Kannan Soundararajan

This note answers a problem posed in Montgomery's CBMS book. The proof was found by ChatGPT 6, and written up by the author.

Primarily human-written textHuman understanding: all partsCollar-Stein LemmaHilbert's inequalitySieves

Sep 28, 2026PDF ↗View work →

hexagon:2609.00102v1

math.NT — Number Theory

Inclusions between p-bounded crystalline loci in dimension two

Contributed by Kalyani Kansal, Brandon Levin, David Savitt

Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion. We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations. We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations).

Primarily human-written textHuman understanding: all parts

Sep 26, 2026PDF ↗View work →