Great problems

Ancient

1. 450 BCE
   Angle trisection with compass and straightedgeGeometry, Algebra. Proved impossible in general by Pierre Wantzel in 1837: trisecting $60^\circ$ requires constructing $\cos(20^\circ)$, whose minimal polynomial $8x^3 - 6x - 1 = 0$ over $\mathbb{Q}$ has degree $3$, which is not a power of $2$.Solved
2. 450 BCE
   Squaring the circleGeometry, Algebra. Squaring the circle with compass and straightedge is impossible, because $\pi$ is transcendental.FLFerdinand von Lindemann, 1882
3. 430 BCE
   Doubling the cubeGeometry, Algebra. Known since antiquity as the Delian problem, doubling the cube by compass and straightedge was proved impossible by Pierre Wantzel in 1837 because $\sqrt[3]{2}$ has algebraic degree $3$ over $\mathbb{Q}$, which is not a power of $2$.Solved
4. 300 BCE
   Constructible regular polygons (Gauss–Wantzel)Geometry, Algebra. By the Gauss–Wantzel theorem (1796–1837), a regular $n$-gon is constructible with compass and straightedge if and only if $n = 2^k p_1 \cdots p_m$ is the product of a power of $2$ and distinct Fermat primes $p_i = 2^{2^{r_i}} + 1$.CGCarl Friedrich Gauss, 1837

Early modern

1. 1611
   Kepler conjectureGeometry. The densest way to stack equal spheres in space is the familiar cannonball pile, with density $\pi/\sqrt{18}$.Hilbert #18THThomas Hales, 1998
2. 1637
   Fermat's Last TheoremArithmetic and number theory. No three positive integers satisfy $x^n + y^n = z^n$ for any integer exponent $n > 2$ — stated by Fermat around 1637 and proved by Andrew Wiles in 1994–1995 (with Richard Taylor) via the modularity of semistable elliptic curves.AWAndrew Wiles, 1994
3. 1650
   Basel problemAnalysis. The sum $\sum_{n=1}^{\infty} 1/n^2$ equals $\pi^2/6$.LELeonhard Euler, 1735

18th century

1. 1736
   Seven Bridges of KönigsbergCombinatorics and discrete mathematics. No such walk exists: the underlying graph has four vertices of odd degree.LELeonhard Euler, 1736
2. 1770
   Insolvability of the quinticAlgebra. Unlike the quadratic, cubic, and quartic formulas, no radical formula solves the general quintic — a fact Galois theory fully explains.NANiels Henrik Abel, 1824
3. 1770
   Waring's problemArithmetic and number theory. Every natural number is the sum of $4$ squares, $9$ cubes, $19$ fourth powers, and so on: conjectured by Edward Waring in 1770 and solved in 1909 when David Hilbert proved that a finite bound $g(k)$ exists for every exponent $k$.DHDavid Hilbert, 1909

19th century

1. 1844
   Catalan's conjectureArithmetic and number theory. 8 and 9 are the only two consecutive perfect powers of positive integers; Catalan conjectured it in 1844, Mihăilescu proved it in 2002.PMPreda Mihăilescu, 2002
2. 1852
   Four colour theoremCombinatorics and discrete mathematics. Every planar map (or loopless planar graph $G$) can be coloured with at most $4$ colours ($\chi(G) \le 4$) — posed by Francis Guthrie in 1852, proved with computer assistance by Kenneth Appel and Wolfgang Haken in 1976, simplified in 1997, and formally verified in Coq by Georges Gonthier in 2005.KAKenneth Appel, 1976

20th century

1. 1900
   Hilbert's seventh problemArithmetic and number theory. Proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider: whenever $\alpha \ne 0, 1$ is algebraic and $\beta$ is an algebraic irrational, $\alpha^\beta$ is transcendental — settling Hilbert's examples $2^{\sqrt{2}}$ and $e^\pi = (-1)^{-i}$ at a single stroke.Hilbert #7Solved
2. 1900
   Hilbert's tenth problemFoundations of mathematics, Arithmetic and number theory. No algorithm can decide, in general, whether a Diophantine equation has an integer solution — Hilbert's tenth problem has a negative answer.Hilbert #10YMYuri Matiyasevich, 1970
3. 1900
   Hilbert's third problemGeometry. Any two polygons of equal area can be cut into finitely many pieces and reassembled into each other, but Max Dehn proved in 1900 that the 3D analogue fails: a cube and a regular tetrahedron of equal volume have different Dehn invariants in $\mathbb{R} \otimes_{\mathbb{Q}} (\mathbb{R}/\pi\mathbb{Q})$ and can never be dissected into one another.Hilbert #3Solved
4. 1900
   Sphere packing in dimensions 8 and 24Geometry. The $E_8$ lattice in $\mathbb{R}^8$ and the Leech lattice $\Lambda_{24}$ in $\mathbb{R}^{24}$ achieve the densest possible sphere packings among all periodic and non-periodic arrangements; solved in March 2016 by Maryna Viazovska (for $n=8$) and Cohn, Kumar, Miller, Radchenko, and Viazovska (for $n=24$).Solved
5. 1904
   Poincaré conjectureTopology, Geometry. Every simply connected closed $3$-manifold is homeomorphic to the $3$-sphere $S^3$ — posed by Henri Poincaré in 1904 and proved by Grigori Perelman in 2002–2003 via Ricci flow with surgery, the only solved Clay Millennium Prize Problem.MillenniumGPGrigori Perelman, 2003
6. 1916
   Bieberbach conjectureAnalysis. Every normalized univalent holomorphic function $f(z) = z + \sum_{n=2}^\infty a_n z^n$ on the open unit disk satisfies $|a_n| \le n$ for all $n \ge 2$ (de Branges's theorem).Solved
7. 1932
   Erdős discrepancy problemCombinatorics and discrete mathematics, Arithmetic and number theory. Every infinite $\pm 1$ sequence has arbitrarily large partial sums along some homogeneous arithmetic progression $d, 2d, \dots, nd$; proved by Terence Tao in 2015.ErdősTTTerence Tao, 2015
8. 1933
   Borsuk's partition problemGeometry. Borsuk's 1933 conjecture that any bounded set in $\mathbb{R}^n$ can be split into $n+1$ parts of smaller diameter holds for $n \le 3$ but was disproved in high dimensions by Jeff Kahn and Gil Kalai in 1993, with counterexamples now known down to $n = 64$ (and a 2026 preprint for $n = 63$).Disproved
9. 1959
   Kadison–Singer problemAnalysis. Every pure state on the diagonal subalgebra $\ell^\infty(\mathbb{N})$ extends uniquely to a pure state on $\mathcal{B}(\ell^2(\mathbb{N}))$, proved in 2013 via interlacing families of polynomials.Solved
10. 1960
    Kervaire invariant one problemTopology. Smooth closed framed manifolds of Kervaire invariant one exist in dimensions $2, 6, 14, 30, 62$, and (per a 2024 preprint) $126$, and in no higher dimensions.Solved
11. 1960
    Sato–Tate conjectureArithmetic and number theory. How do the point counts $\#E(\mathbb{F}_p)$ of a non-CM elliptic curve fluctuate around $p + 1$ as $p$ varies? Formulated independently around 1960 by Mikio Sato and John Tate, the conjecture that the normalized Frobenius angles follow the $\frac{2}{\pi} \sin^2 \theta \, d\theta$ distribution was completely solved between 2008 and 2011 by Clozel, Harris, Shepherd-Barron, Taylor, Barnet-Lamb, and Geraghty.Solved
12. 1963
    Existence of the Lorenz attractorDifferential equations and dynamical systems. Proves that the classical Lorenz differential equations genuinely possess a robust chaotic strange attractor, resolving Smale's 14th problem via a rigorous computer-assisted proof by Warwick Tucker.Smale #14Solved
13. 1963
    Ringel's conjecture (graceful decompositions)Combinatorics and discrete mathematics. Every $n$-edge tree packs $2n+1$ times edge-disjointly into the complete graph $K_{2n+1}$; proved for all sufficiently large $n$ in 2020 by Montgomery, Pokrovskiy, and Sudakov.Solved
14. 1966
    Einstein problem (aperiodic monotile)Geometry. Settled in the affirmative in 2023 when David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss discovered the 13-sided "hat" polykite and the strictly chiral "Spectre" monotile.Solved
15. 1968
    Virtual Haken conjectureTopology. Every closed irreducible 3-manifold with infinite fundamental group has a finite-sheeted cover that contains an embedded incompressible surface.Solved
16. 1972
    Erdős–Faber–Lovász conjectureCombinatorics and discrete mathematics. Can the union of $n$ cliques of size $n$ that pairwise share at most one vertex always be properly vertex-colored with $n$ colors? Posed by Paul Erdős, Vance Faber, and László Lovász in 1972, the conjecture was proved for all sufficiently large $n$ in 2021 by Kang, Kelly, Kühn, Methuku, and Osthus (published in the Annals of Mathematics in 2023), while small cases $n \le 12$ are verified computationally and intermediate values remain open.ErdősSolved
17. 1985
    Bunkbed conjectureCombinatorics and discrete mathematics, Probability and statistics. In percolation on two identical stacked graphs joined by vertical posts, is connection within the same layer always at least as likely as across layers? Posed by Kasteleyn in 1985 and disproved in 2024 by Gladkov, Pak, and Zimin.Disproved
18. 1987
    Fundamental lemma of the Langlands programmeArithmetic and number theory, Algebra. The fundamental lemma is a combinatorial identity between orbital integrals on a reductive group and its endoscopic groups, conjectured by Robert Langlands and Diana Shelstad. It was the last missing ingredient needed to stabilize the Arthur–Selberg trace formula, the main tool linking automorphic representations across different groups. Ngô Bảo Châu proved it in general by reinterpreting the identity geometrically, as a statement about the cohomology of the Hitchin fibration.NCNgô Bảo Châu, 2008
19. 1992
    Sensitivity conjectureCombinatorics and discrete mathematics, Applied and computational mathematics. Is the sensitivity $s(f)$ of a Boolean function polynomially related to its block sensitivity, degree, and decision-tree depth? Posed by Noam Nisan and Mario Szegedy in 1992 and reduced by Gotsman and Linial to a question on hypercube subgraphs, it was solved in a two-page proof by Hao Huang in 2019 showing $\Delta(H) \ge \sqrt{n}$ for any $(2^{n-1} + 1)$-vertex induced subgraph of $Q_n$.Solved
20. 1995
    Cap set problemCombinatorics and discrete mathematics. How large can a subset of $\mathbb{F}_3^n$ be without containing an affine line? Resolved in 2016 when Ellenberg and Gijswijt, adapting the polynomial method of Croot, Lev, and Pach, proved the exponential bound $r_3(\mathbb{F}_3^n) = O(2.756^n)$.Solved