The epistemic state of mathematics' most consequential claims — what is proven, what remains open, what was disproven, and what turned out to be unanswerable from the standard axioms. Every resolved result is graded by verification tier.
Unresolved as of this build. Computational evidence, however extensive, is not proof — see the Mertens Conjecture below for why.
| Problem | Posed | Posed by |
|---|---|---|
| Riemann Hypothesis All non-trivial zeros of the zeta function have real part 1/2. 🏅 Clay Millennium Prize ($1M) — open. Also Hilbert Problem 8. First 10 trillion+ zeros verified computationally to lie on the critical line — strong evidence, not proof |
1859 | Bernhard Riemann |
| Collatz Conjecture Iterating n -> n/2 (even) or 3n+1 (odd) always reaches 1. Verified computationally for all starting values up to ~2^68; Tao (2019) proved 'almost all' orbits reach almost-bounded values |
1937 | Lothar Collatz |
| Goldbach Conjecture (strong form) Every even integer greater than 2 is the sum of two primes. 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. |
1742 | Christian Goldbach (letter to Euler) |
| Twin Prime Conjecture There are infinitely many primes p such that p+2 is also prime. Major progress: Yitang Zhang (2013) proved bounded gaps (70 million); Polymath8b reduced the bound to 246 |
1849 (de Polignac, general form) | Alphonse de Polignac |
| P vs NP Can every problem whose solution is quickly verifiable also be quickly solved? 🏅 Clay Millennium Prize ($1M) — open |
1971 | Stephen Cook (independently Leonid Levin) |
| Navier–Stokes Existence and Smoothness Do smooth solutions to the 3D Navier-Stokes fluid equations always exist for all time? 🏅 Clay Millennium Prize ($1M) — open |
19th century (equations); 2000 (formal problem statement) | Clay Mathematics Institute (formal statement by Fefferman) |
| Hodge Conjecture Certain topological classes on projective algebraic varieties are combinations of algebraic cycle classes. 🏅 Clay Millennium Prize ($1M) — open |
1950 | W.V.D. Hodge |
| Birch and Swinnerton-Dyer Conjecture The rank of an elliptic curve's rational points is determined by the behaviour of its L-function at s=1. 🏅 Clay Millennium Prize ($1M) — open |
1965 | Bryan Birch and Peter Swinnerton-Dyer |
| Yang–Mills Existence and Mass Gap Prove quantum Yang-Mills theory exists rigorously on R^4 and has a positive mass gap. 🏅 Clay Millennium Prize ($1M) — open |
2000 (formal problem statement) | Clay Mathematics Institute (formal statement by Jaffe and Witten) |
| Result | Resolved | By | Verification |
|---|---|---|---|
| Basel Problem Sum of reciprocal squares 1 + 1/4 + 1/9 + ... equals pi^2/6. |
1734 | Leonhard Euler (rigorous proof 1741) | Peer-reviewed |
| Abel–Ruffini Theorem No general algebraic formula (in radicals) exists for polynomial equations of degree 5 or higher. |
1824 | Niels Henrik Abel (Ruffini's 1799 proof was incomplete) | Peer-reviewed |
| Prime Number Theorem The number of primes below x is asymptotically x/ln(x). Formally verified: elementary proof in Isabelle (Avigad et al., 2004); analytic proof in HOL Light (Harrison, 2009) |
1896 | Jacques Hadamard and Charles de la Vallée Poussin (independently) | Formally verified |
| Gödel's Incompleteness Theorems Any consistent formal system containing arithmetic has true statements it cannot prove; no such system can prove its own consistency. Formally verified in multiple systems (e.g. Paulson in Isabelle, 2013; O'Connor in Coq, 2005) |
1931 | Kurt Gödel | Formally verified |
| Nash Equilibrium Existence Theorem Every finite game has at least one equilibrium in mixed strategies. |
1950 | John Nash | Peer-reviewed |
| Four Colour Theorem Every planar map can be coloured with at most four colours so no adjacent regions share a colour. Originally computer-assisted (1976); fully formally verified by Georges Gonthier in Coq (2005) |
1976 | Kenneth Appel and Wolfgang Haken | Formally verified |
| Fermat's Last Theorem No positive integers a,b,c satisfy a^n+b^n=c^n for integer n>2. Full formalization in Lean in progress (project led by Kevin Buzzard, begun 2024) |
1995 | Andrew Wiles (with Richard Taylor) | Peer-reviewed |
| Kepler Conjecture No sphere packing in 3D exceeds density pi/sqrt(18) ~= 74.05% (cannonball stacking is optimal). Originally computer-assisted (1998, published 2005); fully formally verified by the Flyspeck project in HOL Light and Isabelle (2014) |
1998 | Thomas Hales | Formally verified |
| Poincaré Conjecture Every simply connected closed 3-manifold is homeomorphic to the 3-sphere. |
2003 | Grigori Perelman (verification by independent teams completed 2006) | Peer-reviewed |
Conjectures that turned out to be false — including one supported by enormous numerical evidence before its disproof.
| Conjecture | Disproven | By | Method |
|---|---|---|---|
| Mertens Conjecture The Mertens function M(n) is bounded by sqrt(n) in absolute value for all n>1. 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. |
1985 | Andrew Odlyzko and Herman te Riele | Computer-assisted |
| Euler's Sum of Powers Conjecture (entry pending) At least n nth powers are needed to sum to an nth power (generalising Fermat). Disproven by direct computer search: 27^5 + 84^5 + 110^5 + 133^5 = 144^5. The published paper is famously only two sentences long. |
1966 | L.J. Lander and T.R. Parkin | Computer-assisted |
A fourth possibility most people never learn exists: statements that can neither be proven nor disproven from the standard ZFC axioms of mathematics.
| Statement | Established | By | Verification |
|---|---|---|---|
| Continuum Hypothesis There is no set whose cardinality lies strictly between the integers and the real numbers. Neither provable nor disprovable from the standard ZFC axioms — a fundamentally different kind of resolution. Hilbert Problem 1. |
1963 | Kurt Gödel (1940, consistency) + Paul Cohen (1963, independence; Fields Medal 1966) | Peer-reviewed |