{ "registry": "Mathematics Codex — Proof Status Board", "url": "/mathematics/proof-status/", "version": "1.3", "generated": "2026-05-27", "codex_version": "2.5", "status_values": [ "open", "proven", "disproven", "independent" ], "verification_tiers": { "peer-reviewed": "Traditional human peer review of a written proof", "computer-assisted": "Machine computation essential to the proof; programs human-checked", "formally-verified": "Every logical step machine-checked in a proof assistant (Lean/Coq/Isabelle/HOL Light)" }, "results": [ { "id": "fermats-last-theorem", "title": "Fermat's Last Theorem", "statement_short": "No positive integers a,b,c satisfy a^n+b^n=c^n for integer n>2.", "status": "proven", "year_posed": "1637", "posed_by": "Pierre de Fermat", "year_resolved": "1995", "resolved_by": "Andrew Wiles (with Richard Taylor)", "verification": "peer-reviewed", "formalization_notes": "Full formalization in Lean in progress (project led by Kevin Buzzard, begun 2024)", "prize": null, "codex_entry": "/mathematics/named-results/fermats-last-theorem/", "sources": [ "Wiles, A. (1995). Modular elliptic curves and Fermat's Last Theorem. Annals of Mathematics, 141(3), 443-551." ] }, { "id": "poincare-conjecture", "title": "Poincaré Conjecture", "statement_short": "Every simply connected closed 3-manifold is homeomorphic to the 3-sphere.", "status": "proven", "year_posed": "1904", "posed_by": "Henri Poincaré", "year_resolved": "2003", "resolved_by": "Grigori Perelman (verification by independent teams completed 2006)", "verification": "peer-reviewed", "formalization_notes": null, "prize": "Clay Millennium Prize — solved (awarded 2010; declined by Perelman)", "codex_entry": "/mathematics/topology/poincare-conjecture/", "sources": [ "Perelman, G. (2002-2003). arXiv preprints: math/0211159, math/0303109, math/0307245." ] }, { "id": "four-colour-theorem", "title": "Four Colour Theorem", "statement_short": "Every planar map can be coloured with at most four colours so no adjacent regions share a colour.", "status": "proven", "year_posed": "1852", "posed_by": "Francis Guthrie", "year_resolved": "1976", "resolved_by": "Kenneth Appel and Wolfgang Haken", "verification": "formally-verified", "formalization_notes": "Originally computer-assisted (1976); fully formally verified by Georges Gonthier in Coq (2005)", "prize": null, "codex_entry": "/mathematics/discrete/four-colour-theorem/", "sources": [ "Appel, K. and Haken, W. (1977). Every planar map is four colorable. Illinois Journal of Mathematics, 21(3).", "Gonthier, G. (2008). Formal proof — the Four-Color Theorem. Notices of the AMS, 55(11), 1382-1393." ] }, { "id": "kepler-conjecture", "title": "Kepler Conjecture", "statement_short": "No sphere packing in 3D exceeds density pi/sqrt(18) ~= 74.05% (cannonball stacking is optimal).", "status": "proven", "year_posed": "1611", "posed_by": "Johannes Kepler", "year_resolved": "1998", "resolved_by": "Thomas Hales", "verification": "formally-verified", "formalization_notes": "Originally computer-assisted (1998, published 2005); fully formally verified by the Flyspeck project in HOL Light and Isabelle (2014)", "prize": null, "codex_entry": "/mathematics/named-results/kepler-conjecture/", "sources": [ "Hales, T.C. (2005). A proof of the Kepler conjecture. Annals of Mathematics, 162(3), 1065-1185.", "Hales, T. et al. (2017). A formal proof of the Kepler conjecture. Forum of Mathematics, Pi, 5, e2." ] }, { "id": "godels-incompleteness-theorems", "title": "Gödel's Incompleteness Theorems", "statement_short": "Any consistent formal system containing arithmetic has true statements it cannot prove; no such system can prove its own consistency.", "status": "proven", "year_posed": null, "posed_by": null, "year_resolved": "1931", "resolved_by": "Kurt Gödel", "verification": "formally-verified", "formalization_notes": "Formally verified in multiple systems (e.g. Paulson in Isabelle, 2013; O'Connor in Coq, 2005)", "prize": null, "codex_entry": "/mathematics/named-results/godels-incompleteness-theorems/", "sources": [ "Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38, 173-198." ] }, { "id": "basel-problem", "title": "Basel Problem", "statement_short": "Sum of reciprocal squares 1 + 1/4 + 1/9 + ... equals pi^2/6.", "status": "proven", "year_posed": "1650", "posed_by": "Pietro Mengoli", "year_resolved": "1734", "resolved_by": "Leonhard Euler (rigorous proof 1741)", "verification": "peer-reviewed", "formalization_notes": null, "prize": null, "codex_entry": "/mathematics/named-results/basel-problem/", "sources": [ "Euler, L. (1741). De summis serierum reciprocarum. Commentarii academiae scientiarum Petropolitanae, 7, 123-134." ] }, { "id": "abel-ruffini-theorem", "title": "Abel–Ruffini Theorem", "statement_short": "No general algebraic formula (in radicals) exists for polynomial equations of degree 5 or higher.", "status": "proven", "year_posed": null, "posed_by": null, "year_resolved": "1824", "resolved_by": "Niels Henrik Abel (Ruffini's 1799 proof was incomplete)", "verification": "peer-reviewed", "formalization_notes": null, "prize": null, "codex_entry": "/mathematics/algebra/symmetric-group/", "sources": [ "Abel, N.H. (1824). Mémoire sur les équations algébriques." ] }, { "id": "nash-equilibrium-existence", "title": "Nash Equilibrium Existence Theorem", "statement_short": "Every finite game has at least one equilibrium in mixed strategies.", "status": "proven", "year_posed": null, "posed_by": null, "year_resolved": "1950", "resolved_by": "John Nash", "verification": "peer-reviewed", "formalization_notes": null, "prize": null, "codex_entry": "/mathematics/applied/nash-equilibrium/", "sources": [ "Nash, J.F. (1950). Equilibrium points in n-person games. PNAS, 36(1), 48-49." ] }, { "id": "prime-number-theorem", "title": "Prime Number Theorem", "statement_short": "The number of primes below x is asymptotically x/ln(x).", "status": "proven", "year_posed": "1792-1798", "posed_by": "Conjectured by Gauss and Legendre", "year_resolved": "1896", "resolved_by": "Jacques Hadamard and Charles de la Vallée Poussin (independently)", "verification": "formally-verified", "formalization_notes": "Formally verified: elementary proof in Isabelle (Avigad et al., 2004); analytic proof in HOL Light (Harrison, 2009)", "prize": null, "codex_entry": "/mathematics/number-theory/prime-number/", "sources": [ "Hadamard, J. (1896). Sur la distribution des zéros de la fonction zeta(s). Bulletin de la SMF, 24, 199-220." ] }, { "id": "riemann-hypothesis", "title": "Riemann Hypothesis", "statement_short": "All non-trivial zeros of the zeta function have real part 1/2.", "status": "open", "year_posed": "1859", "posed_by": "Bernhard Riemann", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": "First 10 trillion+ zeros verified computationally to lie on the critical line — strong evidence, not proof", "prize": "Clay Millennium Prize ($1M) — open. Also Hilbert Problem 8.", "codex_entry": "/mathematics/named-results/riemann-hypothesis/", "sources": [ "Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie." ] }, { "id": "collatz-conjecture", "title": "Collatz Conjecture", "statement_short": "Iterating n -> n/2 (even) or 3n+1 (odd) always reaches 1.", "status": "open", "year_posed": "1937", "posed_by": "Lothar Collatz", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": "Verified computationally for all starting values up to ~2^68; Tao (2019) proved 'almost all' orbits reach almost-bounded values", "prize": null, "codex_entry": "/mathematics/named-results/collatz-conjecture/", "sources": [ "Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. arXiv:1909.03562." ] }, { "id": "goldbach-conjecture", "title": "Goldbach Conjecture (strong form)", "statement_short": "Every even integer greater than 2 is the sum of two primes.", "status": "open", "year_posed": "1742", "posed_by": "Christian Goldbach (letter to Euler)", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": "Weak (ternary) form — every odd number >5 is a sum of three primes — proven by Harald Helfgott (2013, widely accepted). Strong form verified computationally to 4x10^18.", "prize": null, "codex_entry": "/mathematics/number-theory/goldbach-conjecture/", "sources": [ "Helfgott, H.A. (2013). The ternary Goldbach conjecture is true. arXiv:1312.7748." ] }, { "id": "twin-prime-conjecture", "title": "Twin Prime Conjecture", "statement_short": "There are infinitely many primes p such that p+2 is also prime.", "status": "open", "year_posed": "1849 (de Polignac, general form)", "posed_by": "Alphonse de Polignac", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": "Major progress: Yitang Zhang (2013) proved bounded gaps (70 million); Polymath8b reduced the bound to 246", "prize": null, "codex_entry": "/mathematics/number-theory/twin-prime-conjecture/", "sources": [ "Zhang, Y. (2014). Bounded gaps between primes. Annals of Mathematics, 179(3), 1121-1174." ] }, { "id": "p-vs-np", "title": "P vs NP", "statement_short": "Can every problem whose solution is quickly verifiable also be quickly solved?", "status": "open", "year_posed": "1971", "posed_by": "Stephen Cook (independently Leonid Levin)", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": null, "prize": "Clay Millennium Prize ($1M) — open", "codex_entry": "/mathematics/named-results/p-vs-np/", "sources": [ "Cook, S.A. (1971). The complexity of theorem-proving procedures. Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, 151-158." ] }, { "id": "navier-stokes", "title": "Navier–Stokes Existence and Smoothness", "statement_short": "Do smooth solutions to the 3D Navier-Stokes fluid equations always exist for all time?", "status": "open", "year_posed": "19th century (equations); 2000 (formal problem statement)", "posed_by": "Clay Mathematics Institute (formal statement by Fefferman)", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": null, "prize": "Clay Millennium Prize ($1M) — open", "codex_entry": "/mathematics/mathematical-physics/navier-stokes-existence-and-smoothness/", "sources": [ "Fefferman, C.L. (2000). Existence and smoothness of the Navier-Stokes equation. Clay Mathematics Institute." ] }, { "id": "hodge-conjecture", "title": "Hodge Conjecture", "statement_short": "Certain topological classes on projective algebraic varieties are combinations of algebraic cycle classes.", "status": "open", "year_posed": "1950", "posed_by": "W.V.D. Hodge", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": null, "prize": "Clay Millennium Prize ($1M) — open", "codex_entry": "/mathematics/named-results/clay-millennium-problems/", "sources": [ "Hodge, W.V.D. (1950). The topological invariants of algebraic varieties. Proceedings of the ICM, Cambridge, MA." ] }, { "id": "birch-swinnerton-dyer", "title": "Birch and Swinnerton-Dyer Conjecture", "statement_short": "The rank of an elliptic curve's rational points is determined by the behaviour of its L-function at s=1.", "status": "open", "year_posed": "1965", "posed_by": "Bryan Birch and Peter Swinnerton-Dyer", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": null, "prize": "Clay Millennium Prize ($1M) — open", "codex_entry": "/mathematics/named-results/clay-millennium-problems/", "sources": [ "Birch, B.J. and Swinnerton-Dyer, H.P.F. (1965). Notes on elliptic curves. II. Journal für die reine und angewandte Mathematik, 218, 79-108." ] }, { "id": "yang-mills-mass-gap", "title": "Yang–Mills Existence and Mass Gap", "statement_short": "Prove quantum Yang-Mills theory exists rigorously on R^4 and has a positive mass gap.", "status": "open", "year_posed": "2000 (formal problem statement)", "posed_by": "Clay Mathematics Institute (formal statement by Jaffe and Witten)", "year_resolved": null, "resolved_by": null, "verification": null, "formalization_notes": null, "prize": "Clay Millennium Prize ($1M) — open", "codex_entry": "/mathematics/named-results/clay-millennium-problems/", "sources": [ "Jaffe, A. and Witten, E. (2000). Quantum Yang-Mills theory. Clay Mathematics Institute." ] }, { "id": "continuum-hypothesis", "title": "Continuum Hypothesis", "statement_short": "There is no set whose cardinality lies strictly between the integers and the real numbers.", "status": "independent", "year_posed": "1878", "posed_by": "Georg Cantor", "year_resolved": "1963", "resolved_by": "Kurt Gödel (1940, consistency) + Paul Cohen (1963, independence; Fields Medal 1966)", "verification": "peer-reviewed", "formalization_notes": "Neither provable nor disprovable from the standard ZFC axioms — a fundamentally different kind of resolution. Hilbert Problem 1.", "prize": null, "codex_entry": "/mathematics/foundations-logic/continuum-hypothesis/", "sources": [ "Cohen, P.J. (1963). The independence of the continuum hypothesis. PNAS, 50(6), 1143-1148.", "Gödel, K. (1940). The Consistency of the Continuum Hypothesis. Princeton University Press." ] }, { "id": "mertens-conjecture", "title": "Mertens Conjecture", "statement_short": "The Mertens function M(n) is bounded by sqrt(n) in absolute value for all n>1.", "status": "disproven", "year_posed": "1897", "posed_by": "Franz Mertens (earlier Stieltjes, 1885)", "year_resolved": "1985", "resolved_by": "Andrew Odlyzko and Herman te Riele", "verification": "computer-assisted", "formalization_notes": "Disproven via computational analysis of zeta zeros — no explicit counterexample n is known even today. A cautionary tale: extensive numerical evidence suggested it was true, and its truth would have implied the Riemann Hypothesis.", "prize": null, "codex_entry": "/mathematics/named-results/mertens-conjecture/", "sources": [ "Odlyzko, A.M. and te Riele, H.J.J. (1985). Disproof of the Mertens conjecture. Journal für die reine und angewandte Mathematik, 357, 138-160." ] }, { "id": "eulers-sum-of-powers-conjecture", "title": "Euler's Sum of Powers Conjecture", "statement_short": "At least n nth powers are needed to sum to an nth power (generalising Fermat).", "status": "disproven", "year_posed": "1769", "posed_by": "Leonhard Euler", "year_resolved": "1966", "resolved_by": "L.J. Lander and T.R. Parkin", "verification": "computer-assisted", "formalization_notes": "Disproven by direct computer search: 27^5 + 84^5 + 110^5 + 133^5 = 144^5. The published paper is famously only two sentences long.", "prize": null, "codex_entry": "/mathematics/named-results/eulers-sum-of-powers-conjecture/", "sources": [ "Lander, L.J. and Parkin, T.R. (1966). Counterexample to Euler's conjecture on sums of like powers. Bulletin of the AMS, 72(6), 1079." ] } ], "last_updated": "2026-07-09" }