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. At this conference he presented 10 of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) and the list was published later.

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.

The 24th problem

In preparing the problems Hilbert had 24 problems listed, but decided against one of the problems. The 24th problem was in proof theory on a criterion for simplicity and general methods. Discovery of this problem is due to Rüdiger Thiele.

External links

• Listing of the 23 problems, with descriptions of which have been solved
• English translation of Hilbert's 1900 address
• Details on the solution of the 18th problem
• The Mathematical Gazette, March 2000 (page 2-8) "100 Years On"
• "On Hilbert's 24th Problem: Report on a New Source and Some Remarks."

References

• Rowe, David; Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford University Press. ISBN 0198506511

• Yandell, Benjamin H. (2002). The Honors Class. Hilbert's Problems and Their Solvers. A K Peters. ISBN 1568811411