Open problem in mathematics

Unsolved problems in mathematics

This article lists some unsolved problems in mathematics. See individual articles for details and sources.

Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six ones yet to be solved are:

Only the Poincaré conjecture has been solved. The smooth four dimensional Poincaré conjecture is still unsolved. That is, can a four dimensional topological sphere have two or more inequivalent smooth structures?

Other still-unsolved problems

Additive number theory

Number theory: prime numbers

General number theory

Discrete geometry

Ramsey theory

General algebra


  • Number of Magic squares
  • Finding a formula for the probability that two elements chosen at random generate the symmetric group S_n
  • Frankl's union-closed sets conjecture that any family of sets closed under unions has an element contained in half or more of the sets
  • The Lonely runner conjecture: if k runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be more than a distance 1/(k+1) from each other runner) at some time?

Graph theory


Group theory


Problems solved recently

See also


Books discussing unsolved problems

  • Fan Chung; Ron Graham (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 1-56881-111-X.
  • Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3.
  • Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer. ISBN 0-387-20860-7.
  • Victor Klee; Stan Wagon (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0-88385-315-9.
  • Marcus Du Sautoy (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0060935588.
  • Keith Devlin (2006). The Millennium Problems - The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 0-7607-8659-8.

Books discussing recently solved problems

  • Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1841157910.
  • Donal O'Shea (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-846-14012-9.
  • George G. Szpiro (2003). Kepler's Conjecture. Wiley. ISBN 0-471-08601-0.
  • Mark Ronan (2006). Symmetry and the Monster. Oxford. ISBN 0-19-280722-6.


  • Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site

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