# Open problems from the 1950s > 23 AI-made results on questions first raised the 1950s, each explained in plain English. Results on questions first raised the 1950s, going by the… Canonical: https://closedform.nishachay.tech/open-since/1950s/ Last updated: 2026-10-14 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](https://closedform.nishachay.tech/e/openai-math-2026-025/). Results rated Landmark include [The rational Hodge conjecture for CM abelian varieties](https://closedform.nishachay.tech/e/openai-math-2026-032/) and [Nagata’s conjecture and maximal Seshadri constants](https://closedform.nishachay.tech/e/openai-math-2026-039/). ## Results (23) - [Prime-factor statistics of p-1](https://closedform.nishachay.tech/e/openai-math-2026-011/) - Entry 011 · The prime factors of p − 1, for a random prime p, are claimed to split like a randomly broken stick. (Claimed, not checked) - [Ostmann’s inverse Goldbach conjecture](https://closedform.nishachay.tech/e/openai-math-2026-013/) - Entry 013 · The primes cannot be built by adding two sets of whole numbers. (Formal proof listed) - [The irrationality exponent of π is 2](https://closedform.nishachay.tech/e/openai-math-2026-017/) - Entry 017 · Fractions can never get unusually close to π. (Formal proof listed) - [Squarefree quartics and power-free polynomial values](https://closedform.nishachay.tech/e/openai-math-2026-020/) - Entry 020 · Whole-number quartics like n⁴ + 2 take squarefree values at exactly the share the local guess predicts. (Formal proof listed) - [Short Egyptian fractions](https://closedform.nishachay.tech/e/openai-math-2026-025/) - Entry 025 · Every fraction below 1 is a sum of about log log b distinct unit fractions, as Erdős conjectured. (Formal proof listed) - [The rational Hodge conjecture for CM abelian varieties](https://closedform.nishachay.tech/e/openai-math-2026-032/) - Entry 032 · The rational Hodge conjecture is claimed for every abelian variety with complex multiplication, in every dimension. (Claimed, not checked) - [Nagata’s conjecture and maximal Seshadri constants](https://closedform.nishachay.tech/e/openai-math-2026-039/) - Entry 039 · Nagata's 1959 rule for curves through random points holds for every count from 10 points up. (Formal proof listed) - [Real ultraflat Littlewood polynomials and unbounded binary merit factors](https://closedform.nishachay.tech/e/openai-math-2026-076/) - Entry 076 · Polynomials with only +1 and −1 coefficients can be almost perfectly flat on the unit circle. (Formal proof listed) - [Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy](https://closedform.nishachay.tech/e/openai-math-2026-090/) - Entry 090 · No arrangement of points in the plane beats the triangular lattice for any steadily fading repulsion. (Formal proof listed) - [The optimal order of convex-body covering density](https://closedform.nishachay.tech/e/openai-math-2026-092/) - Entry 092 · The worst convex shape needs covering density about n log n in dimension n, lattice or not. (Formal proof listed) - [Hyperbolicity cones without semidefinite lifts](https://closedform.nishachay.tech/e/openai-math-2026-095/) - Entry 095 · A claimed proof that some hyperbolicity cones have no semidefinite description, even with hidden variables. (Formal proof listed) - [Cylinder coverings below the half-area bound](https://closedform.nishachay.tech/e/openai-math-2026-100/) - Entry 100 · Cylinders can cover a regular tetrahedron for less than half its smallest shadow. (Formal proof listed) - [Existential–universal real sentences in the counting hierarchy](https://closedform.nishachay.tech/e/openai-math-2026-141/) - Entry 141 · Deciding “there is x so that for every y” questions about real numbers claimed at one fixed counting level. (Claimed, not checked) - [The Euclidean plane cannot be colored with five colors](https://closedform.nishachay.tech/e/openai-math-2026-158/) - Entry 158 · No way of coloring the plane with five colors keeps every unit-distance pair apart. (Formal proof listed) - [Foulkes' conjecture for sixth powers and quadratic stabilization](https://closedform.nishachay.tech/e/openai-math-2026-210/) - Entry 210 · Foulkes' conjecture is claimed for sixth powers, with a Lean-proved bound past which a natural map is onto. (Formal proof listed) - [A finitely generated Eilenberg–Ganea counterexample](https://closedform.nishachay.tech/e/openai-math-2026-249/) - Entry 249 · Claims a group of cohomological dimension two with no two-dimensional classifying space, refuting Eilenberg–Ganea. (Formal proof listed) - [Amenability, unitarizability, and strong Ulam stability](https://closedform.nishachay.tech/e/openai-math-2026-251/) - Entry 251 · A discrete group is amenable exactly when every uniformly bounded representation can be made unitary. (Formal proof listed) - [Localization and delocalization in the Anderson model](https://closedform.nishachay.tech/e/openai-math-2026-261/) - Entry 261 · Claims electrons stay trapped on a random square grid at any disorder, and can spread at weak disorder in 3D. (Formal proof listed) - [Kadison's similarity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-288/) - Entry 288 · Every bounded map from a C*-algebra to operators can be straightened to respect the mirror operation. (Formal proof listed) - [Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness](https://closedform.nishachay.tech/e/openai-math-2026-294/) - Entry 294 · Proves Kaplansky's quasitrace conjecture false, and that stable finiteness can fail for tensor products. (Formal proof listed) - [Approximation and quadratic strong-operator paving](https://closedform.nishachay.tech/e/openai-math-2026-300/) - Entry 300 · Claims to settle both paving conjectures of Popa and Vaes for every maximal abelian subalgebra of a von Neumann algebra. (Claimed, not checked) - [Cyclic length and chromatic fixed-point loss](https://closedform.nishachay.tech/e/openai-math-2026-314/) - Entry 314 · Claims the exact chromatic loss between fixed points of p-group actions is a count of cyclic steps. (Claimed, not checked) - [Nonhomeomorphic closed aspherical four-manifolds](https://closedform.nishachay.tech/e/openai-math-2026-320/) - Entry 320 · A paper claims two closed aspherical 4-manifolds with the same homotopy type but no homeomorphism, against Borel. 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