Category:Hilbert 23

The Hilbert 23 is a list of twenty-three problems in mathematics published by David Hilbert during 1900.

They are as follows:

1. The Continuum Hypothesis.
2. Proof that the axioms of mathematics are consistent.
3. Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?
4. Construct all metrics where lines are geodesics.
5. Are continuous groups automatically differential groups?
6. Axiomatize all of physics.
7. Is $a^b$ transcendental, for algebraic $a \ne 0, 1$ and irrational algebraic $b$? (Proved affirmative by the Gelfond-Schneider Theorem.)
8. The Riemann Hypothesis and the Goldbach Conjecture.
9. Find the most general law of the Reciprocity Theorem in any algebraic number field.
10. Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.
11. Solving quadratic forms with algebraic numerical coefficients.
12. Extend the Kronecker-Weber Theorem on abelian extensions of the rational numbers to any base number field.
13. Solve all 7th degree equations using functions of two parameters.
14. Proof of the finiteness of certain complete systems of functions.
15. Rigorous foundation of Schubert's Enumerative Calculus.
16. Topology of algebraic curves and surfaces.
17. Expression of definite rational function as quotient of sums of squares.
18. Is there a non-regular, space-filling polyhedron? What is the densest Sphere Packing?
19. Are the solutions of Lagrangians always analytic?
20. Do all variational problems with certain boundary conditions have solutions?
21. Proof of the existence of linear differential equations having a prescribed monodromic group.
22. Uniformization of analytic relations by means of automorphic functions.
23. Further development of the calculus of variations.

There was originally going to be a 24th problem, on a criterion for simplicity and general methods in proof theory, but Hilbert decided not to include it, as it was (like numbers 4, 6, 16 and 23) too vague to ever be described as "solved".