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## Encyclopedia > Hilbert's third problem

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# Hilbert's problems

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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 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". He lists the fourth problem as too vague to say whether it has been solved. He also lists the 18th problem as "open" in his 2000 book, because the sphere-packing problem 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, and progress continues. 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.

• Listing of the 23 problems, with descriptions of which have been solved Prime Math Encyclopedia (http://www.mathacademy.com/pr/prime/articles/hilbert_prob/index.asp?PRE=hilber&TAL=Y&TAN=Y&TBI=Y&TCA=Y&TCS=Y&TEC=Y&TFO=Y&TGE=Y&TNT=Y&TPH=Y&TST=Y&TTO=Y&TTR=Y&TAD=)
• Jeremy Gray, 2000, The Hilbert Challenge, ISBN 0198506511
• The solution of the 18th problem was found in 2000 according to this (http://www.math.pitt.edu/articles/hilbert)

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