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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 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 non-mathematical Axiomatize all of physics
Problem 7 solved Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic 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 list 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.
Rowe, David; Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford University Press. ISBN 0198506511
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