Hilbert's problems

In 1900, the mathematician David Hilbert published a list of 23 unsolved mathematical problems. The list of problems turned out to be very influential.

After Hilbert's death, another problem was found in his writings; this is sometimes known as Hilbert's 24th problem today. This problem is about finding criteria to show that a solution to a problem is the simplest possible.

Of the 23 problems, three were unresolved in 2012, three were too vague to be resolved, and six could be partially solved. Given the influence of the problems, the Clay Mathematics Institute formulated a similar list, called the Millennium Prize Problems in  2000.

## Overview

The formulation of certain problems is better than that of others.
Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19, 20, and 21 have a resolution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 15, 18<sup>+</sup>, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether it resolves the problem.

The solution for problem 18, the Kepler conjecture, uses a computer-assisted proof.  This is controversial, because a human reader is unable to verify the proof in reasonable time.

That leaves 16, 8 – the Riemann hypothesis – and 12 unresolved. On this classification 4, 16, and 23 are too vague to ever be described as solved. The withdrawn 24 would also be in this class. 6 is considered as a problem in physics rather than in mathematics.

## Table of problems

Hilbert's twenty-three problems are:
|width=4%|Problem|width=25%|Brief explanation|width=254%|Status|width=4%|Year Solved|
| --- | --- | --- | --- | --- | --- | --- | --- |
|1st|The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers)||1963|||||
|2nd|Prove that the axioms of arithmetic are consistent.||1936?|||||
|3rd|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?||1900|||||
|4th|Construct all metrics where lines are geodesics.||–|||||
|5th|Are continuous groups automatically differential groups?||1953?|||||
|6th|Axiomatize all of physics||–|||||
|7th|Is _a_<sup>_ b_</sup> transcendental, for algebraic _a_ ≠ 0,1 and irrational algebraic _b_ ?||1934|||||
|8th|The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is ½") and other prime number problems, among them Goldbach's conjecture and the twin prime conjecture||–|||||
|9th|Find most general law of the reciprocity theorem in any algebraic number field||–|||||
|10th|Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.||1970|||||
|11th|Solving quadratic forms with algebraic numerical coefficients.||–|||||
|12th|Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.||–|||||
|13th|Solving 7-th degree equations using continuous functions of two parameters.||1957|||||
|14th|Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated?||1959|||||
|15th|Rigorous foundation of Schubert's enumerative calculus.||–|||||
|16th|Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane.||–|||||
|17th|Expression of definite rational function as quotient of sums of squares||1927|||||
|18th|(a) Is there a polyhedron which admits only an anisohedral tiling in three dimensions? (b) What is the densest sphere packing?||(a) 1928 (b) 1998|||||
|19th|Are the solutions of Lagrangians always analytic?||1957|||||
|20th|Do all variational problems with certain boundary conditions have solutions?||–|||||
|21st|Proof of the existence of linear differential equations having a prescribed monodromic group||–|||||
|22nd|Uniformization of analytic relations by means of automorphic functions||–|||||
|23rd|Further development of the calculus of variations||–|||||

## Other websites

<div style="margin:0.1em 0 0;">  <span class="plainlinks">**:Wikilivres:Mathematische Probleme** – source texts, documents, translations, media</span></div>
- Listing of the 23 problems, with descriptions of which have been solved
- Original text of Hilbert's talk, in German
- English translation of Hilbert's _Mathematical Problems_ on Wikisource
- Details on the solution of the 18th problem
- "On Hilbert's 24th Problem: Report on a New Source and Some Remarks."
- The Paris Problems
- Hilbert's Tenth Problem page!
- 'From Hilbert's Problems to the Future' , lecture by Professor Robin Wilson, Gresham College, 27 February 2008 (available in text, audio and video formats).