Open problems from the 1950s
Results on questions first raised the 1950s, going by the earliest step each entry records.
23 AI-made results on questions first raised the 1950s, from 47 papers in the OpenAI math collection, each explained in plain English. 17 have a formal Lean proof listed; 6 are claimed in a paper and not yet machine-checked. They span 11 fields, led by number theory (5), convex and metric geometry (4) and operator algebras (3). The oldest question here dates back to 1950: Short Egyptian fractions. Results rated Landmark include The rational Hodge conjecture for CM abelian varieties and Nagata’s conjecture and maximal Seshadri constants.
Number theory5
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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.
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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.
Formal proof listed -
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.
Formal proof listed -
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.
Formal proof listed -
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.
Formal proof listed
Algebraic and complex geometry2
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The rational Hodge conjecture for CM abelian varieties
Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.
Claimed, not checked -
Nagata’s conjecture and maximal Seshadri constants
Proves Nagata's strict inequality for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic.
Formal proof listed
Real and complex analysis1
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Real ultraflat Littlewood polynomials and unbounded binary merit factors
Constructs polynomials with N consecutive coefficients in whose modulus is uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.
Formal proof listed
Convex and metric geometry4
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Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy
Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.
Formal proof listed -
The optimal order of convex-body covering density
Determines the optimal worst-case covering density as , for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most ; centrally symmetric examples in every sufficiently large dimension require at least even without the lattice restriction, for absolute .
Formal proof listed -
Hyperbolicity cones without semidefinite lifts
Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.
Formal proof listed -
Cylinder coverings below the half-area bound
Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.
Formal proof listed
Theoretical computer science1
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Existential–universal real sentences in the counting hierarchy
Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.
Claimed, not checked
Combinatorics1
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The Euclidean plane cannot be colored with five colors
Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.
Formal proof listed
Algebra1
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Foulkes' conjecture for sixth powers and quadratic stabilization
Proves the sixth case of Foulkes’ conjecture: embeds equivariantly in for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map is surjective for a ≥ 2 and , giving dimension-independent quadratic stabilization.
Formal proof listed
Group theory2
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A finitely generated Eilenberg–Ganea counterexample
Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.
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Amenability, unitarizability, and strong Ulam stability
Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.
Formal proof listed
Mathematical physics1
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Localization and delocalization in the Anderson model
Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.
Formal proof listed
Operator algebras3
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Kadison's similarity conjecture
Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a -homomorphism after conjugation by a bounded invertible operator.
Formal proof listed -
Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor .
Formal proof listed -
Approximation and quadratic strong-operator paving
Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.
Claimed, not checked
Topology2
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Cyclic length and chromatic fixed-point loss
Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.
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Nonhomeomorphic closed aspherical four-manifolds
Constructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.
Claimed, not checked