# List of unsolved problems in mathematics

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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, six have yet to be solved:

The seventh problem, 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 (sequence A006052 in OEIS)
• Finding a formula for the probability that two elements chosen at random generate the symmetric group $S_n$
• Frankl's union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
• The Lonely runner conjecture: if $k+1$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance $1/(k+1)$ from each other runner) at some time?
• Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
• The 1/3–2/3 conjecture: does every finite partially ordered set contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?
• Conway's thrackle conjecture

### Discrete geometry

• Solving the Happy Ending problem for arbitrary $n$
• Finding matching upper and lower bounds for K-sets and halving lines
• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies

### Dynamical system

• Furstenberg conjecture – Is every invariant and ergodic measure for the $\times 2,\times 3$ action on the circle either Lebesgue or atomic?
• Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups

## References

1. ^
2. ^
3. ^
4. ^
5. ^
6. ^
7. ^ An introduction to irrationality and transcendence methods
8. ^ Some unsolved problems in number theory
9. ^ Ribenboim, P. (2006). Die Welt der Primzahlen (in German) (2 ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
10. ^ Malliaris, M.; Shelah, S. (2012), Cofinality spectrum theorems in model theory, set theory and general topology, arXiv:1208.5424
11. ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society 125: 1503–1509 Unknown parameter |Article Stable URL= ignored (help).
12. ^ Franciscos Santos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics (Princeton University and Institute for Advanced Study) 176 (1): 383–412. doi:10.4007/annals.2012.176.1.7.
13. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (I)", Inventiones Mathematicae 178 (3): 485–504, doi:10.1007/s00222-009-0205-7 and Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (II)", Inventiones Mathematicae 178 (3): 505–586, doi:10.1007/s00222-009-0206-6.
14. ^ Green, Ben (2004), "The Cameron–Erdős conjecture", The Bulletin of the London Mathematical Society 36 (6): 769–778, arXiv:math.NT/0304058, doi:10.1112/S0024609304003650, MR 2083752.

### 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 0-06-093558-8.
• John Derbyshire (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0-309-08549-7.
• Keith Devlin (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN [[Special:BookSources/0-7607-8659-8|0-7607-8659-8[[Category:Articles with invalid ISBNs]]]] Check |isbn= value (help).
• Vincent D. Blondel, Alexandre Megrestski (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0-691-11748-9.