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	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.

External links

• 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=)
• A transcript of Hilbert's 1900 address, with a useful table of contents, and a listing of books addressing each question, is available at http://aleph0.clarku.edu/~djoyce/hilbert/toc (http://aleph0.clarku.edu/~djoyce/hilbert/toc)
• 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)