A comprehensive reference tracking the status, researchers, and AI involvement in major open mathematical conjectures — from the Riemann Hypothesis to recent AI breakthroughs. Updated April 2026. Created with Claude.
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Key: Solved AI Solo solve AI Collaborative Disputed
Major Conjectures
| Problem | Field | Year | Status | AI Involvement |
|---|---|---|---|---|
| Riemann Hypothesis | Analytic Number Theory | 1859 | Unsolved | Assistive only (Ramanujan Machine; LLMs) |
| Birch & Swinnerton-Dyer Conjecture | Number Theory / Arithmetic Geometry | 1965 | Unsolved | None reported |
| Hodge Conjecture | Algebraic Geometry / Topology | 1950 | Unsolved | None reported |
| Navier–Stokes Existence & Smoothness | PDEs / Fluid Dynamics | 1845 / 2000 | Unsolved | None reported |
| P versus NP | Theoretical Computer Science | 1971 | Unsolved | None reported |
| Yang–Mills Existence & Mass Gap | Mathematical Physics | 1954 / 2000 | Unsolved | None reported |
| Poincaré Conjecture (3-D) | Topology | 1904 | Solved (2003) | None |
| Smooth 4-D Poincaré Conjecture | Topology / 4-Manifold Theory | ≈1960 | Unsolved | None reported |
| Twin Prime Conjecture | Number Theory | 1849 | Unsolved (bounded gaps proven) | Assistive (LLMs) |
| Goldbach Conjecture | Number Theory | 1742 | Unsolved | None reported |
| Collatz Conjecture (3n+1) | Dynamical Systems | 1937 | Unsolved | None reported |
| abc Conjecture | Number Theory / Arithmetic Geometry | 1985 | Disputed | None reported |
| Beal Conjecture | Number Theory | 1993 | Unsolved | None reported |
| Hadwiger–Nelson Problem | Combinatorial Geometry | 1950 | Partial: 5 ≤ χ(R²) ≤ 7 | Computational (SAT solvers; ML) |
| Cap Set Problem | Extremal Combinatorics | Mid-20th c. | Active progress | AI co-discovery (FunSearch, DeepMind 2023) |
| Sphere Packing (dims 8 & 24) | Geometry / Number Theory | 1611+ | Solved in dims 1,2,3,8,24 | Computational/formal (Lean Flyspeck) |
| Kissing Number Problem (dim 11) | Discrete Geometry | 1694 | Partial (open in most dims) | AI co-discovery (AlphaEvolve, 2025) |
| Erdős Discrepancy Problem | Combinatorics / Number Theory | 1932 | Solved (2015) | Computational pre-proof (SAT solver) |
| Geometric Langlands Conjecture | Representation Theory | 1980s | Solved (2024) | None |
| Unique Games Conjecture | Computational Complexity | 2002 | Unsolved | None reported |
| Hadwiger Conjecture (graph theory) | Graph Theory | 1943 | Unsolved | None reported |
| Erdős Conjecture on Arithmetic Progressions | Combinatorial Number Theory | 1936 | Unsolved | None reported |
AI Wins in Mathematics, 2023–2026
| Problem | Date | AI System | Type | Achievement |
|---|---|---|---|---|
| Cap Set Problem (low-dim) | Dec 2023 | FunSearch (DeepMind) | AI co-discovery | Largest cap sets in dim 8 — biggest improvement in 20 years. Published in Nature. |
| IMO 2024 (4/6 problems) | July 2024 | AlphaProof + AlphaGeometry 2 | AI Solo (with Lean) | Silver-medal threshold (28 points). First AI medal-level IMO performance. |
| Matrix Multiplication (4×4 complex) | May 2025 | AlphaEvolve (DeepMind) | AI Solo | First improvement on Strassen (1969): 48 multiplications instead of 49. |
| Kissing Number (dim 11) | May 2025 | AlphaEvolve (DeepMind) | AI co-discovery | Improved lower bound on a problem dating to Newton–Gregory (1694). |
| IMO 2025 (5/6 problems) | July 2025 | Gemini Deep Think | AI Solo | Gold-medal threshold (35 pts). First AI officially graded by IMO coordinators. |
| Erdős Problem #846 | Oct–Nov 2025 | GPT-5 (OpenAI) | AI co-discovery | First instance Sawhney described as genuinely AI-generated mathematics. |
| Erdős Problem #728 | Jan 2026 | GPT-5.2 Pro + Aristotle | AI Solo (autonomous) | First Erdős problem regarded as fully resolved autonomously by AI. Lean-verified by Tao. |
| Chen–Gendron Conjecture | Feb 2026 | AxiProver (Axiom Math) | AI Solo | Open since 2021 — resolved from natural language, formalized in Lean. |
| ~100 Erdős Problems (total) | Oct 2025–Apr 2026 | GPT-5/5.2, Gemini, Claude, AlphaProof, Aristotle | Mixed | ~100 problems moved to ‘solved’ on Tao’s tracker. A growing tail are genuinely new proofs. |
Sources: Clay Mathematics Institute; erdosproblems.com; github.com/teorth/erdosproblems; Nature (Nov 2025); DeepMind blog; Terence Tao’s blog. Compiled April 28, 2026.