Great problems

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

19th century

1. 1859
   Riemann hypothesisArithmetic and number theory, Analysis. The zeta function encodes the distribution of prime numbers: its zeros control the error term in the prime counting function $\pi(x)$. Riemann observed in 1859 that all the nontrivial zeros he could compute lay on the critical line $\mathrm{Re}(s)=\tfrac12$ and conjectured this always holds. A proof would give the strongest possible form of the prime number theorem; billions of zeros have been checked numerically and all lie on the line, but no proof or disproof is known. It is one of the seven Clay Millennium Prize Problems.MillenniumHilbert #8Open
2. 1878
   Continuum hypothesisFoundations of mathematics. Cantor showed that the reals are uncountable, so their cardinality $2^{\aleph_0}$ is strictly larger than that of the integers, $\aleph_0$. The continuum hypothesis (CH) asks whether $2^{\aleph_0}$ is the very next cardinal, $\aleph_1$. Gödel (1940) showed CH cannot be disproved from the standard ZFC axioms, and Cohen (1963) showed it cannot be proved either, using the new method of forcing. CH is therefore independent of ZFC: both CH and its negation are consistent with the usual axioms of set theory, so the question has no answer within ZFC alone.Hilbert #1KGKurt Gödel, 1963

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 sixteenth problemDifferential equations and dynamical systems, Geometry. Asks for the possible arrangements of ovals of a degree-$n$ real algebraic curve and for a uniform bound $H(n)$ on the number of limit cycles of a degree-$n$ polynomial vector field in the plane. Every individual polynomial system has finitely many limit cycles (Ilyashenko 1991; Écalle 1992), but whether $H(n) < \infty$ remains open even for $n = 2$.Hilbert #16Smale #13Open
3. 1900
   Hilbert's sixth problemMathematical physics. Hilbert asked both for an axiomatic foundation of probability and mechanics and for a rigorous derivation of macroscopic fluid equations from microscopic particle collisions. Kolmogorov axiomatized probability in 1933, and in 2025 Deng, Hani, and Ma derived the long-time Boltzmann and fluid equations from hard-sphere dynamics, while axiomatizing all of modern physics remains open.Hilbert #6Partially solved
4. 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
5. 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
6. 1900
   Hilbert's twelfth problemArithmetic and number theory. Asks for explicit analytic generators of every abelian extension of a number field $K$, generalizing how roots of unity $\exp(2\pi i/n)$ generate $\mathbb{Q}^{\mathrm{ab}}$. Solved for imaginary quadratic fields by complex multiplication and for totally real fields in a $p$-adic sense by Dasgupta and Kakde (2021–2023), while the general complex-analytic problem remains open.Hilbert #12Open