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

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