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# Unsolved problems in mathematics

This article lists some . 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

### Combinatorics

• 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 $1/\left(k+1\right)$ from each other runner) at some time?

## 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.

## Resources

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

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