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Hilbert's problems

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 solvedThe continuum hypothesis
Problem 2 solvedAre the axioms of arithmetic consistent?
Problem 3 solvedCan two tetrahedra be proved to have equal volume (under certain assumptions)?
Problem 4 too vague Construct all metrics where lines are geodesics
Problem 5 solvedAre continuous groups automatically differential groups?
Problem 6 openAxiomatize all of physics
Problem 7 partially solvedIs ab transcendental, for algebraic a ≠ 0,1 and irrational b?
Problem 8 open The Riemann hypothesis and Goldbach's conjecture
Problem 9 solvedFind most general law of reciprocity in any algebraic number field
Problem 10 solvedDetermination of the solvability of a diophantine equation
Problem 11 solvedQuadratic forms with algebraic numerical coefficients
Problem 12 solvedAlgebraic number field extensions
Problem 13 solvedSolve all 7-th degree equations using functions of two arguments
Problem 14 solvedProof of the finiteness of certain complete systems of functions
Problem 15 solvedRigorous foundation of Schubert's enumerative calculus
Problem 16 open Topology of algebraic curves and surfaces
Problem 17 solvedExpression of definite rational function as quotient of sums of squares
Problem 18 solvedIs there a non-regular, space-filling polyhedron? What's the densest sphere packing?
Problem 19 solvedAre the solutions of Lagrangians always analytic?
Problem 20 solvedDo all variational problems with certain boundary conditions have solutions?
Problem 21 solvedProof of the existence of linear differential equations having a prescribed monodromic group
Problem 22solvedUniformization of analytic relations by means of automorphic functions
Problem 23 solvedFurther 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.

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