Open problems from the 1940s
Results on questions first raised the 1940s, going by the earliest step each entry records.
12 AI-made results on questions first raised the 1940s, from 16 papers in the OpenAI math collection, each explained in plain English. 7 have a formal Lean proof listed; 5 are claimed in a paper and not yet machine-checked. They span 6 fields, led by combinatorics (3), dynamical systems and ergodic theory (3) and algebra (2). The oldest question here dates back to 1940: A counterexample to Kaplansky’s zero-divisor conjecture. Results rated Landmark include Hilbert’s tenth problem over ℚ and Rokhlin’s multiple-mixing problem.
Number theory2
-
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.
Claimed, not checked -
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.
Claimed, not checked
Dynamical systems and ergodic theory3
-
Banach’s simple Lebesgue-spectrum problem
Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L2 space: the bilateral iterates of one real observable form an orthonormal basis of that space.
Formal proof listed -
Rokhlin’s multiple-mixing problem
Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge.
Formal proof listed -
The entropy-rate dimension formula for self-similar measures
For every self-similar measure on the line generated by finitely many contracting similarities, proves , where is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.
Formal proof listed
Combinatorics3
-
Graph coloring, clique minors, and Colin de Verdière invariants
Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy , where is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound . In the positive direction, every finite nonempty graph satisfies for a universal constant C.
Formal proof listed -
The higher-dimensional Erdős distinct-distances conjecture
For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least distinct distances, with depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
Claimed, not checked -
Planar distinct distances and unit-distance bounds
Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but points of any n-point planar set determine at least distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by for an absolute δ > 0.
Formal proof listed
Algebra2
-
A counterexample to Kaplansky’s zero-divisor conjecture
Constructs a finitely presented torsion-free group G whose group algebra has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.
Formal proof listed -
A counterexample to Kurosh’s division-ring problem
Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.
Claimed, not checked
Probability and statistical mechanics1
-
The joint critical Ashkin–Teller current limit
Identifies the joint scaling limit of Ashkin–Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible lattice approximations and wired primal/free dual boundaries, the height converges to the predicted Gaussian free field, and the clusters to canonical recursive sets of that same field, retaining every nesting depth.
Claimed, not checked
Operator algebras1
-
Isomorphism of the free group factors
Resolves the free group factor isomorphism problem: , and hence all interpolated free group factors, including , are isomorphic. Their common factor has fundamental group .
Formal proof listed