Hilbert's problems are a list of 23 problems in mathematics put forth by David Hilbert in the Paris conference of the International Congress of Mathematicians in 1900. The problems were all unsolved at the time, and several of them turned out to be very influential for twentieth-century mathematics.
Hilbert's 23 problems are:
|Problem 1||solved||The continuum hypothesis|
|Problem 2||solved||Are the axioms of arithmetic consistent?|
|Problem 3||solved||Can two tetrahedra be proved to have equal volume (under certain assumptions)?|
|Problem 4||too vague||Construct all metrics where lines are geodesics|
|Problem 5||solved||Are continuous groups automatically differential groups?|
|Problem 6||open||Axiomatize all of physics|
|Problem 7||partially solved||Is ab transcendental, for algebraic a ≠ 0,1 and irrational b?|
|Problem 8||open||The Riemann hypothesis and Goldbach's conjecture|
|Problem 9||solved||Find most general law of reciprocity in any algebraic number field|
|Problem 10||solved||Determination of the solvability of a diophantine equation|
|Problem 11||solved||Quadratic forms with algebraic numerical coefficients|
|Problem 12||solved||Algebraic number field extensions|
|Problem 13||solved||Solve all 7-th degree equations using functions of two arguments|
|Problem 14||solved||Proof of the finiteness of certain complete systems of functions|
|Problem 15||solved||Rigorous foundation of Schubert's enumerative calculus|
|Problem 16||open||Topology of algebraic curves and surfaces|
|Problem 17||solved||Expression of definite rational function as quotient of sums of squares|
|Problem 18||solved||Is there a non-regular, space-filling polyhedron? What's the densest sphere packing?|
|Problem 19||solved||Are the solutions of Lagrangians always analytic?|
|Problem 20||solved||Do all variational problems with certain boundary conditions have solutions?|
|Problem 21||solved||Proof of the existence of linear differential equations having a prescribed monodromic group|
|Problem 22||solved||Uniformization of analytic relations by means of automorphic functions|
|Problem 23||solved||Further development of the calculus of variations|
According to Rowe & Gray (see reference below), most of the problems have been solved. Some were not completely defined, but enough progress has been made to consider them "solved"; Rowe & Gray lists the fourth problem as too vague to say whether it has been solved.
They also lists the 18th problem as "open" in their 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below). Advances were made on problem 16 as recently as the 1990s.
Problem 8 contains two famous problems, both of which remain unsolved. The first of them, the Riemann hypothesis, is one of the seven Millennium Prize Problems, which were intended to be the "Hilbert Problems" of the 21st century.