# Open problems from the 1970s

> 63 AI-made results on questions first raised the 1970s, each explained in plain English. Results on questions first raised the 1970s, going by the…

Canonical: https://closedform.nishachay.tech/open-since/1970s/
Last updated: 2026-10-14

Results on questions first raised the 1970s, going by the earliest step each entry records.

63 AI-made results on questions first raised the 1970s, from 132 papers in the OpenAI math collection, each explained in plain English. 39 have a formal Lean proof listed; 24 are claimed in a paper and not yet machine-checked. They span 17 fields, led by algebraic and complex geometry (7), combinatorics (6) and theoretical computer science (6). The oldest question here dates back to 1970: [Deterministic polynomial factorization over prime fields](https://closedform.nishachay.tech/e/openai-math-2026-142/). Results rated Landmark include [Kakeya in three and four dimensions](https://closedform.nishachay.tech/e/openai-math-2026-074/) and [Fourier restriction for positively curved surfaces](https://closedform.nishachay.tech/e/openai-math-2026-077/).

## Results (63)

- [Goldfeld’s conjecture: densities and mean analytic rank](https://closedform.nishachay.tech/e/openai-math-2026-006/) - Entry 006 · Among the quadratic twists of any elliptic curve, half are claimed to have analytic rank 0 and half rank 1. (Claimed, not checked)
- [Independent largest prime factors of consecutive integers](https://closedform.nishachay.tech/e/openai-math-2026-012/) - Entry 012 · The largest prime factors of n and n + 1 act like independent dice, so each wins half the time. (Formal proof listed)
- [The Margulis–Platonov conjecture over global fields](https://closedform.nishachay.tech/e/openai-math-2026-018/) - Entry 018 · Normal subgroups of these simple groups over any global field are claimed to come from finitely many local pieces. (Claimed, not checked)
- [Patterson's first moment for cubic Gauss sums](https://closedform.nishachay.tech/e/openai-math-2026-023/) - Entry 023 · Cubic Gauss sums over primes lean positive at exactly the rate Patterson predicted in 1978. (Formal proof listed)
- [An asymptotic formula for the number of totients](https://closedform.nishachay.tech/e/openai-math-2026-024/) - Entry 024 · Counting totient values up to x gets an exact leading formula, and doubling x doubles the count in the limit. (Formal proof listed)
- [Iitaka subadditivity, variation, and logarithmic additivity](https://closedform.nishachay.tech/e/openai-math-2026-033/) - Entry 033 · Twisting a family of shapes never destroys the forms its fibers and base contribute. (Formal proof listed)
- [Bloch’s conjecture for complex surfaces](https://closedform.nishachay.tech/e/openai-math-2026-040/) - Entry 040 · On a complex surface with no holomorphic two-forms, zero-cycles carry only their degree and Albanese data. (Claimed, not checked)
- [Shafarevich counterexamples in dimension two and with large fundamental group](https://closedform.nishachay.tech/e/openai-math-2026-046/) - Entry 046 · Two claimed counterexamples to Shafarevich's conjecture: a surface and a fourfold whose universal covers fail convexity. (Claimed, not checked)
- [Zariski cancellation and affine fibrations over the complex numbers](https://closedform.nishachay.tech/e/openai-math-2026-047/) - Entry 047 · A four-dimensional shape that is not ordinary space, until you add one line. (Formal proof listed)
- [Counterexamples to Zariski’s multiplicity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-059/) - Entry 059 · Two hypersurface germs with the same embedded topology but different multiplicities, against Zariski’s 1971 question. (Claimed, not checked)
- [The Global Spherical Shell conjecture](https://closedform.nishachay.tech/e/openai-math-2026-060/) - Entry 060 · Every minimal class VII surface with positive second Betti number contains a global spherical shell. (Claimed, not checked)
- [Topological triviality of μ -constant surface singularities](https://closedform.nishachay.tech/e/openai-math-2026-064/) - Entry 064 · A claimed proof that surface singularities with a constant Milnor number keep the same topology along a family. (Claimed, not checked)
- [Brennan's conjecture and the integral-means spectrum](https://closedform.nishachay.tech/e/openai-math-2026-072/) - Entry 072 · Reshaping a region into a disk can never squeeze too hard: Brennan's 1978 conjecture is proved. (Formal proof listed)
- [Kakeya in three and four dimensions](https://closedform.nishachay.tech/e/openai-math-2026-074/) - Entry 074 · Claimed proofs: the Kakeya maximal conjecture in 3D, and full dimension for every Kakeya set in 4D. (Claimed, not checked)
- [Fourier restriction for positively curved surfaces](https://closedform.nishachay.tech/e/openai-math-2026-077/) - Entry 077 · Claimed proofs of the full Fourier restriction range p > 3 for the sphere and for positively curved surfaces. (Claimed, not checked)
- [The geometric case of the Erdős similarity conjecture](https://closedform.nishachay.tech/e/openai-math-2026-084/) - Entry 084 · A set covering almost all of [0, 1] can dodge every copy of a geometric list like 1/2, 1/4, 1/8, … (Formal proof listed)
- [Sharp projection-body inequalities and a counterexample to simplex maximization](https://closedform.nishachay.tech/e/openai-math-2026-088/) - Entry 088 · In four or more dimensions, ellipsoids have the smallest shadow body for their volume. (Formal proof listed)
- [Exact derandomization of logarithmic space: L=RL=BPL](https://closedform.nishachay.tech/e/openai-math-2026-103/) - Entry 103 · Coin flips give no extra power to computers with tiny memory: L = RL = BPL. (Formal proof listed)
- [Almost-linear approximation of edit distance](https://closedform.nishachay.tech/e/openai-math-2026-121/) - Entry 121 · Edit distance can be estimated to within any fixed accuracy in barely more than linear time. (Formal proof listed)
- [Polynomial-time scheduling on three identical machines](https://closedform.nishachay.tech/e/openai-math-2026-124/) - Entry 124 · The best schedule for one-hour jobs with order rules on three machines can be found in polynomial time. (Formal proof listed)
- [Exponential state costs for two-way automata](https://closedform.nishachay.tech/e/openai-math-2026-129/) - Entry 129 · Removing guesses from simple two-way reading machines can cost exponentially many states. (Formal proof listed)
- [Subset Sum in O(2^(0.49n)) time](https://closedform.nishachay.tech/e/openai-math-2026-138/) - Entry 138 · A randomized algorithm claims to decide every Subset Sum instance in O(2^(0.49n)) time, below the 2^(n/2) scale. (Claimed, not checked)
- [Deterministic polynomial factorization over prime fields](https://closedform.nishachay.tech/e/openai-math-2026-142/) - Entry 142 · A claimed coin-free algorithm that factors any polynomial modulo a prime in time polynomial in the input size. (Claimed, not checked)
- [A C 1 counterexample to the entropy conjecture](https://closedform.nishachay.tech/e/openai-math-2026-151/) - Entry 151 · A smooth-enough map can double a shape's homology while its orbits stay orderly. (Formal proof listed)
- [Zero entropy does not guarantee a smooth positive-volume model](https://closedform.nishachay.tech/e/openai-math-2026-152/) - Entry 152 · An abstract zero-entropy system with no smooth, volume-keeping copy in any dimension. (Formal proof listed)
- [Counterexamples to Ryser’s covering conjecture](https://closedform.nishachay.tech/e/openai-math-2026-162/) - Entry 162 · Ryser's covering conjecture is false: some intersecting hypergraphs with r parts need r vertices to cover. (Formal proof listed)
- [Hindman’s finite sums and products conjecture](https://closedform.nishachay.tech/e/openai-math-2026-164/) - Entry 164 · However whole numbers are colored, some set of each size has its sums and products in one color, the paper claims. (Claimed, not checked)
- [The hypercube Ramsey conjecture](https://closedform.nishachay.tech/e/openai-math-2026-171/) - Entry 171 · Two-coloring the links among a constant times 2ⁿ points always gives a one-color n-dimensional cube, the paper claims. (Claimed, not checked)
- [Classification of finite Euclidean Ramsey configurations](https://closedform.nishachay.tech/e/openai-math-2026-172/) - Entry 172 · An exact algebraic test tells which finite point sets must appear in one color in any coloring of space. (Formal proof listed)
- [Power savings for intersective polynomial differences and prime arguments](https://closedform.nishachay.tech/e/openai-math-2026-182/) - Entry 182 · Sets of whole numbers with no square difference are smaller than N by a fixed power of N. (Formal proof listed)
- [Power savings for planar halving lines and k -sets](https://closedform.nishachay.tech/e/openai-math-2026-183/) - Entry 183 · The maximum number of halving lines for n points in the plane is below n to the power 4/3 by a fixed margin. (Formal proof listed)
- [A counterexample to the small Cohen–Macaulay module conjecture](https://closedform.nishachay.tech/e/openai-math-2026-195/) - Entry 195 · A three-dimensional complete local domain over the complex numbers has no small Cohen–Macaulay module. (Formal proof listed)
- [A torsion-free group algebra that is not directly finite](https://closedform.nishachay.tech/e/openai-math-2026-197/) - Entry 197 · In some group algebras, a left inverse is not a right inverse, which disproves Kaplansky's direct-finiteness conjecture. (Formal proof listed)
- [Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures](https://closedform.nishachay.tech/e/openai-math-2026-199/) - Entry 199 · A module with no self-extensions in any degree can still fail to be projective, against Auslander–Reiten and Tachikawa. (Formal proof listed)
- [The ℓ¹-Bass conjecture for all discrete groups](https://closedform.nishachay.tech/e/openai-math-2026-207/) - Entry 207 · Idempotents in group rings carry no trace at infinite-order elements, for every group; Kaplansky's problem follows. (Formal proof listed)
- [Integral counterexamples to Gersten’s conjecture](https://closedform.nishachay.tech/e/openai-math-2026-209/) - Entry 209 · Two ramified rings are claimed to break Gersten's conjecture: nonzero K-theory classes vanish over their fractions. (Claimed, not checked)
- [Canonical O(3) continuum limit and exact O(4) mass asymptotics](https://closedform.nishachay.tech/e/openai-math-2026-215/) - Entry 215 · Spin models with three or more components are proved to lose alignment exponentially at every temperature. (Formal proof listed)
- [Critical and near-critical XY scaling and BKT universality](https://closedform.nishachay.tech/e/openai-math-2026-216/) - Entry 216 · Six papers claim exact critical laws for the planar XY model, as the Berezinskii–Kosterlitz–Thouless picture predicts. (Claimed, not checked)
- [Critical SK autocorrelation processes and dynamics across the temperature transition](https://closedform.nishachay.tech/e/openai-math-2026-227/) - Entry 227 · Heat-bath mixing in the SK spin glass is fast below β = 1, takes n^(2/3) at β = 1, and slows sharply above it. (Formal proof listed)
- [Continuum phase transitions for radial pair potentials](https://closedform.nishachay.tech/e/openai-math-2026-228/) - Entry 228 · Two built pair potentials give particles in space a first-order phase transition across a range of densities. (Formal proof listed)
- [All-temperature pressure of orthogonally invariant Ising spin glasses](https://closedform.nishachay.tech/e/openai-math-2026-234/) - Entry 234 · A formula for the pressure of orthogonally invariant Ising spin glasses holds at every temperature. (Formal proof listed)
- [Single-fold Diophantine representations and undecidability under an at-most-one-solution promise](https://closedform.nishachay.tech/e/openai-math-2026-242/) - Entry 242 · Every computably listable set gets a whole-number polynomial test with exactly one solution per member. (Formal proof listed)
- [Thompson's group F is nonamenable](https://closedform.nishachay.tech/e/openai-math-2026-248/) - Entry 248 · Thompson's group F, a famous group of interval stretchings, is not amenable. (Formal proof listed)
- [Boone–Higman embeddings with higher finiteness](https://closedform.nishachay.tech/e/openai-math-2026-250/) - Entry 250 · A group has a word-solving algorithm exactly when it sits inside a finitely presented simple group. (Formal proof listed)
- [Classifying spaces and geometric obstructions for Artin groups](https://closedform.nishachay.tech/e/openai-math-2026-254/) - Entry 254 · Proves the Artin K(π,1) conjecture in finite rank, and finds an Artin group with no geometric CAT(0) action. (Formal proof listed)
- [Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions](https://closedform.nishachay.tech/e/openai-math-2026-260/) - Entry 260 · Claims the spacetime Penrose inequality: a black hole's enclosing area caps how small the total mass can be. (Formal proof listed)
- [Sharp finite-matrix Lieb–Thirring inequalities and all equality cases](https://closedform.nishachay.tech/e/openai-math-2026-262/) - Entry 262 · Proves the sharp 1D Lieb–Thirring constant for 1/2 < γ < 3/2; claims the matrix case and all equality cases. (Formal proof listed)
- [Strong cosmic censorship near two-ended Kerr data](https://closedform.nishachay.tech/e/openai-math-2026-264/) - Entry 264 · Claims typical vacuum perturbations of a rotating Kerr black hole cannot be smoothly continued past the inner horizon. (Claimed, not checked)
- [Strong Kadison–Kastler stability and its spatial boundaries](https://closedform.nishachay.tech/e/openai-math-2026-289/) - Entry 289 · Proves, with a Lean check, that close von Neumann algebras are conjugate by a unitary near the identity, uniformly. (Formal proof listed)
- [Invariant projections, hyperinvariant subspaces, and transitive algebras](https://closedform.nishachay.tech/e/openai-math-2026-293/) - Entry 293 · Proves that some operators on Hilbert space have no hyperinvariant subspace, a negative answer to a long-open problem. (Formal proof listed)
- [Finite Smith–Toda complexes at every height](https://closedform.nishachay.tech/e/openai-math-2026-308/) - Entry 308 · Claims Smith–Toda complexes exist at every height if the prime may grow, with an explicit V(4) at the prime 1009. (Claimed, not checked)
- [The Kervaire invariant problem at the prime three](https://closedform.nishachay.tech/e/openai-math-2026-309/) - Entry 309 · Claims to settle which odd-primary Kervaire classes survive at the prime three: stems 10, 106 and 322. (Claimed, not checked)
- [Quillen's conjecture in rational homology](https://closedform.nishachay.tech/e/openai-math-2026-310/) - Entry 310 · Claims Quillen's 1978 conjecture on subgroup posets for every finite group and every prime, in a stronger homology form. (Claimed, not checked)
- [The four-dimensional Singer conjecture](https://closedform.nishachay.tech/e/openai-math-2026-315/) - Entry 315 · Claims the Singer conjecture in dimension four: L²-Betti numbers of aspherical four-manifolds vanish off degree two. (Claimed, not checked)
- [Curtis’s conjecture](https://closedform.nishachay.tech/e/openai-math-2026-316/) - Entry 316 · A paper claims to prove Curtis's 1975 conjecture: only η, ν, σ and Kervaire classes show up in mod-2 sphere homology. (Claimed, not checked)
- [Lipschitz equivalent Banach spaces need not be linearly isomorphic](https://closedform.nishachay.tech/e/openai-math-2026-324/) - Entry 324 · Two separable Banach spaces match distances up to fixed factors but are not linearly the same, answering Kalton. (Formal proof listed)
- [A counterexample to metric-entropy duality](https://closedform.nishachay.tech/e/openai-math-2026-329/) - Entry 329 · Proves that the duality conjecture for metric entropy is false, already when the covering body is a cube. (Formal proof listed)
- [A smooth surface metric with no local isometric immersion in ℝ 3](https://closedform.nishachay.tech/e/openai-math-2026-334/) - Entry 334 · Some smooth rules for measuring length cannot be built as a surface in space, even near one point. (Formal proof listed)
- [Sharp singular-set bounds for stationary integral varifolds](https://closedform.nishachay.tech/e/openai-math-2026-346/) - Entry 346 · Claims that the singular points of a stationary integral varifold have dimension at most one less than the varifold. (Claimed, not checked)
- [Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample](https://closedform.nishachay.tech/e/openai-math-2026-353/) - Entry 353 · Proves affine Bernstein rigidity for complete graphs in dimensions 3 to 9; claims a smooth dimension-ten counterexample. (Formal proof listed)
- [Kinetic limits and fluctuations over the Boltzmann lifespan](https://closedform.nishachay.tech/e/openai-math-2026-364/) - Entry 364 · Claims the Boltzmann equation follows from Newton's laws for short-range forces for as long as it stays regular. (Claimed, not checked)
- [The hot spots conjecture for simply connected planar domains](https://closedform.nishachay.tech/e/openai-math-2026-369/) - Entry 369 · On smooth flat shapes without holes, the lasting hot and cold spots lie on the edge. (Formal proof listed)
- [De Giorgi's conjecture in dimension eight](https://closedform.nishachay.tech/e/openai-math-2026-375/) - Entry 375 · Claims De Giorgi's conjecture in dimension eight: monotone Allen–Cahn solutions there are flat one-dimensional profiles. (Claimed, not checked)

## Related

- [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results
- [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results
- [Named conjectures](https://closedform.nishachay.tech/topic/named-conjectures/) - Topic · 85 results
- [Landmark results](https://closedform.nishachay.tech/significance/landmark/) - Significance · 49 results
- [Open since before 1940](https://closedform.nishachay.tech/open-since/before-1940/) - Era · 16 results
- [Open since the 1940s](https://closedform.nishachay.tech/open-since/1940s/) - Era · 12 results
- [Open since the 1950s](https://closedform.nishachay.tech/open-since/1950s/) - Era · 23 results
- [Open since the 1960s](https://closedform.nishachay.tech/open-since/1960s/) - Era · 33 results
- [Open since the 1980s](https://closedform.nishachay.tech/open-since/1980s/) - Era · 60 results
- [Open since the 1990s](https://closedform.nishachay.tech/open-since/1990s/) - Era · 55 results
- [Algebraic and complex geometry](https://closedform.nishachay.tech/science/mathematics/algebraic-and-complex-geometry/) - Field · 7 here
- [Theoretical computer science](https://closedform.nishachay.tech/science/mathematics/theoretical-computer-science/) - Field · 6 here
- [Combinatorics](https://closedform.nishachay.tech/science/mathematics/combinatorics/) - Field · 6 here
- [All topics](https://closedform.nishachay.tech/topics/) - Every way to browse