Open problems from the 1960s
Results on questions first raised the 1960s, going by the earliest step each entry records.
33 AI-made results on questions first raised the 1960s, from 50 papers in the OpenAI math collection, each explained in plain English. 25 have a formal Lean proof listed; 8 are claimed in a paper and not yet machine-checked. They span 15 fields, led by theoretical computer science (6), number theory (5) and combinatorics (4). The oldest question here dates back to 1960: A quadratic bound for Jacobsthal’s function. Results rated Landmark include The full BSD formula from low Selmer corank and The L log L Fourier-convergence conjecture.
Number theory5
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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.
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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 .
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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.
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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.
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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.
Formal proof listed
Algebraic and complex geometry2
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A characteristic-zero counterexample to Lipman–Zariski
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
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A counterexample to Griffiths’ positivity conjecture
Constructs ample rank-two bundles on with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.
Formal proof listed
Real and complex analysis1
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The Fourier-convergence conjecture
Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the scale.
Formal proof listed
Theoretical computer science6
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Matrix multiplication with exponent at most 9/4
Proves over ℂ, giving arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.
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Integer multiplication below
Multiplies two n-bit integers exactly at every input length in deterministic worst-case time , with , on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen optimality conjecture in the ordinary multitape bit model.
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Almost-linear-time exact matching and prescribed-degree factors in general graphs
Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.
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Exact Fourier transforms below
Gives a deterministic length-n discrete Fourier transform algorithm using operations for every n, with explicit . The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.
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The computational complexity of Weisfeiler–Leman refinement
Proves unconditional deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete.
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Generalized star height at most three
Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.
Formal proof listed
Combinatorics4
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The Harary–Hill and Zarankiewicz crossing-number formulas
Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs.
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The circulant Hadamard and Barker-sequence conjectures
Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when .
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Barnette’s Hamiltonian-cycle conjecture
Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.
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The Erdős–Gallai cycle-decomposition conjecture
Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.
Formal proof listed
Algebra3
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Serre’s intersection-multiplicity conjecture
Proves strict positivity of Serre's intersection multiplicity for nonzero finitely generated modules over any regular local ring, provided has finite length and . This resolves the positivity conjecture, including ramified mixed characteristic.
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Lech’s multiplicity conjecture
Proves for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.
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A counterexample to finitistic-dimension finiteness
Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.
Formal proof listed
Probability and statistical mechanics2
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Planar first-passage geometry and the absence of bigeodesics
Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C1 boundary. Differentiability also holds for every Gamma law with positive shape and rate.
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Exact three- and four-state reconstruction thresholds and four-state tree capacity
Proves the exact reconstruction threshold , with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d > 1 and d > 0, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of λ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.
Formal proof listed
Mathematical logic1
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Shelah's eventual categoricity and the prescribed-threshold obstruction
Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice.
Formal proof listed
Group theory1
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Nonsingular systems of equations over arbitrary groups
Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.
Formal proof listed
Mathematical physics1
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Failure of Kohn–Sham ensemble representation
Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in . This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.
Claimed, not checked
Operator algebras2
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The Kadison–Ringrose cohomology conjecture
Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.
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The generator problem for finite factors
Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion of such factors, the unitaries u with form a dense Gδ subset in the trace 2-norm topology.
Formal proof listed
Topology1
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A counterexample to Wall's finite D(2) problem
Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.
Claimed, not checked
Functional analysis2
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Independence of the separable quotient problem
Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields.
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Nonexpansive fixed points in reflexive Banach spaces
Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity.
Formal proof listed
Differential geometry1
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Negative Kähler curvature without bounded holomorphic coordinates
Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.
Formal proof listed
Partial differential equations1
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Interior regularity for infinity-harmonic functions
Proves uniform interior regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.
Claimed, not checked