Number theory
31 families · 17 verified by lean
This page gathers the 31 result families that the openai/math repository catalogues under Number theory. Each claim comes from a manuscript; 17 of them have a Lean formalization checked by software, and the other 14 are still to be verified. Open any card to read a plain-language explainer, the original statement and its sources.
Milne’s rationality conjecture and algebraic specialization
Proves Milne's rationality conjecture for abelian varieties over 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.
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.
The quasi-Riemann hypothesis
Proves that every Dirichlet L-function, including , is zero-free in , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over . A companion gives a different proof of the zero-free half-plane .
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.
Irrationality of Catalan’s constant
Proves that Catalan's constant is irrational.
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.
Ordinary two-point correlations and the corrected Elliott conjecture
Proves the ordinary two-point Chowla conjecture, with a bound 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 for .
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.
Function-field reconstruction from Milnor K-theory and Galois data
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from , , 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.
Unrestricted pro-modularity at the prime two
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of 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.
Prime-factor statistics of
Proves that the normalized ordered logarithms of the prime factors of , counted with multiplicity, converge jointly to the Poisson–Dirichlet law as p ranges uniformly over primes up to x and . This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than totient preimages, for every ε > 0.
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 are asymptotically independent in ordinary natural density. In particular, the integers satisfying have density 1/2.
Ostmann’s inverse Goldbach conjecture
Proves that no finite modification of the primes can be written as with each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.
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 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 . 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.
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.
Zilber–Pink in abelian varieties and the Siegel threefold
Proves the abelian Zilber–Pink conjecture over : 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 for Hodge-generic curves defined over , without boundary or reduction assumptions.
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 satisfies . This also proves convergence of the Flint–Hills series , with angles in radians.
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 is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.
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 ℚp 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 and of genus at least two.
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 -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.
A quadratic bound for Jacobsthal’s function
Answers Jacobsthal's quadratic-bound question: every interval of 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.
The weak inhomogeneous Duffin–Schaeffer conjecture
Proves that for every real shift γ and finite-valued , divergence of implies 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.
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 . Every fixed nonzero prime-angle Fourier mode has smaller order.
An asymptotic formula for the number of totients
Gives an asymptotic equivalent for the number of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, for every fixed c > 0, answering Erdős and Hall’s scaling question.
Short Egyptian fractions
Every rational with is a sum of distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.
Positive lower density of large prime gaps
For every fixed C > 0, a positive proportion of consecutive prime gaps exceed , throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where increases have positive lower density, answering Erdős and Prachar.
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 SLr character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.
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.
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 primes in every sufficiently large interval have primitive root a, with .
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.
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.