Subjects

Number theory

31 results from OpenAI’s math release. 31 explained so far. OpenAI has published Lean proofs for 17: 14 of the main theorem, 3 of part of the claim.

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• 001 Milne’s rationality conjecture and algebraic specialization

  Proves Milne's rationality conjecture for abelian varieties over $\overline{\mathbb Q}$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.

  ▶Episode, 6:09

• 002 The full BSD formula from low Selmer corank

  Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.

  3 papers ▶Episode, 8:10

• 003 The quasi-Riemann hypothesis

  Proves that every Dirichlet L-function, including $\zeta(s)$, is zero-free in $\Re s\gt 7/8$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $\mathbb Q(\sqrt{-3})$. A companion gives a different proof of the zero-free half-plane $\Re s\gt 11/12$.

  3 papers ▶Episode, 8:45 OpenAI published a Lean proof of the main theorem.

• 004 Hilbert’s tenth problem over ℚ

  Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.

  2 papers ▶Episode, 6:50

• 005 Irrationality of Catalan’s constant

  Proves that Catalan's constant $G=\sum_{j\ge0}(-1)^j/(2j+1)^2$ is irrational.

  ▶Episode, 7:02 OpenAI published a Lean proof of the main theorem.

• 006 Goldfeld’s conjecture: densities and mean analytic rank

  Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.

  2 papers ▶Episode, 6:38

• 007 Ordinary two-point correlations and the corrected Elliott conjecture

  Proves the ordinary two-point Chowla conjecture, with a bound $O(X/(\log X)^c)$ for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $n^{it}$ for $|t|\le X$.

  ▶Episode, 7:51 OpenAI published a Lean proof of the main theorem.

• 008 The Deligne–Drinfeld conjecture

  Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture.

  ▶Episode, 6:27 OpenAI published a Lean proof of the main theorem.

• 009 Function-field reconstruction from Milnor K-theory and Galois data

  Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $K^{\mathrm M}_1/\ell$, $K^{\mathrm M}_2/\ell$, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.

  3 papers ▶Episode, 6:03 OpenAI published a Lean proof of part of the claim.

• 010 Unrestricted pro-modularity at the prime two

  Every continuous odd absolutely irreducible two-dimensional 2-adic representation of $G_{\mathbb Q}$ unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.

  3 papers ▶Episode, 5:19

• 011 Prime-factor statistics of $p-1$

  Proves that the normalized ordered logarithms of the prime factors of $p-1$, counted with multiplicity, converge jointly to the Poisson–Dirichlet law $\mathrm{PD}(1)$ as p ranges uniformly over primes up to x and $x\to\infty$. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than $n^{1-\varepsilon}$ totient preimages, for every ε > 0.

  3 papers ▶Episode, 6:11

• 012 Independent largest prime factors of consecutive integers

  Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $n+1$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $P^+(n)\lt P^+(n+1)$ have density 1/2.

  ▶Episode, 5:15 OpenAI published a Lean proof of the main theorem.

• 013 Ostmann’s inverse Goldbach conjecture

  Proves that no finite modification of the primes can be written as $A+B$ with $A,B\subseteq\mathbb Z_{\ge0}$ each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.

  ▶Episode, 4:23 OpenAI published a Lean proof of the main theorem.

• 014 Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

  Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over $\overline{\mathbb F}_q$ under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and $p\nmid |W_G|$. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.

  8 papers ▶Episode, 5:29

• 015 Torus-packet equidistribution in prime, quartic, and sextic degrees

  Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.

  3 papers ▶Episode, 5:27 OpenAI published a Lean proof of the main theorem.

• 016 Zilber–Pink in abelian varieties and the Siegel threefold

  Proves the abelian Zilber–Pink conjecture over $\overline{\mathbb Q}$: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold $\mathcal A_2$ for Hodge-generic curves defined over $\overline{\mathbb Q}$, without boundary or reduction assumptions.

  4 papers ▶Episode, 5:08

• 017 The irrationality exponent of π is 2

  Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational $p/q$ satisfies $|\pi-p/q|\ge q^{-2-\varepsilon}$. This also proves convergence of the Flint–Hills series $\sum_{n\ge1}1/(n^3\sin^2 n)$, with angles in radians.

  ▶Episode, 4:37 OpenAI published a Lean proof of the main theorem.

• 018 The Margulis–Platonov conjecture over global fields

  Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of $G(k)$ is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.

  2 papers ▶Episode, 6:57

• 019 The local p-adic section conjecture and global consequences

  Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚ_p correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves $X_0(N)$ and $X_1(N)$ of genus at least two.

  2 papers ▶Episode, 5:51

• 020 Squarefree quartics and power-free polynomial values

  Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the $(d-2)$-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.

  ▶Episode, 4:54 OpenAI published a Lean proof of the main theorem.

• 021 A quadratic bound for Jacobsthal’s function

  Answers Jacobsthal's quadratic-bound question: every interval of $Ck^2$ consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.

  ▶Episode, 4:36 OpenAI published a Lean proof of the main theorem.

• 022 The weak inhomogeneous Duffin–Schaeffer conjecture

  Proves that for every real shift γ and finite-valued $\psi:\mathbb N\to[0,\infty)$, divergence of $\sum_q\phi(q)\psi(q)/q$ implies $\|qx-\gamma\|\lt \psi(q)$ for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.

  ▶Episode, 4:33

• 023 Patterson's first moment for cubic Gauss sums

  Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order $X^{5/6}/\log X$. Every fixed nonzero prime-angle Fourier mode has smaller order.

  ▶Episode, 5:14 OpenAI published a Lean proof of the main theorem.

• 024 An asymptotic formula for the number of totients

  Gives an asymptotic equivalent for the number $V(x)$ of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $V(cx)/V(x)\to c$ for every fixed c > 0, answering Erdős and Hall’s scaling question.

  ▶Episode, 4:20 OpenAI published a Lean proof of the main theorem.

• 025 Short Egyptian fractions

  Every rational $a/b$ with $1\le a\lt b$ is a sum of $O(\log\log b)$ distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.

  ▶Episode, 4:22 OpenAI published a Lean proof of the main theorem.

• 026 Positive lower density of large prime gaps

  For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $C\log p_n$, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where $p_n/n$ increases have positive lower density, answering Erdős and Prachar.

  ▶Episode, 3:47 OpenAI published a Lean proof of part of the claim.

• 027 Potential integral density on curve character varieties

  Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SL_r character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.

  ▶Episode, 6:18 OpenAI published a Lean proof of part of the claim.

• 028 Uniformly bounded components of Gaussian-prime graphs

  Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes.

  ▶Episode, 4:31 OpenAI published a Lean proof of the main theorem.

• 029 Primitive roots for every admissible integer base

  Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least $c_a x/(\log x)^2$ primes in every sufficiently large interval $(x,2x)$ have primitive root a, with $c_a\gt 0$.

  2 papers ▶Episode, 5:11

• 030 Modularity of elliptic curves over imaginary quadratic fields

  Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.

  ▶Episode, 4:53

• 031 Uchida’s conjecture for open homomorphisms of Galois groups

  Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.

  ▶Episode, 4:17

Summary and paper titles from OpenAI’s catalog, used under the Apache License 2.0. Last checked October 19, 2026.