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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, the six ones yet to be solved are:## Other still-unsolved problems

### Additive number theory

### Number theory: prime numbers

### General number theory

### Discrete geometry

### Ramsey theory

### General algebra

### Combinatorics

### Graph theory

### Analysis

### Group theory

### Other

## Problems solved recently

## See also

## References

## Books discussing unsolved problems

## Books discussing recently solved problems

## Resources

- P versus NP
- The Hodge conjecture
- The Riemann hypothesis
- Yang-Mills existence and mass gap
- Navier-Stokes existence and smoothness
- The Birch and Swinnerton-Dyer conjecture

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?

- Goldbach's conjecture and its weak version
- The values of $g(k)$ and $G(k)$ in Waring's problem
- Collatz conjecture ($3n\; +\; 1$ conjecture)
- Gilbreath's conjecture

- Catalan's Mersenne conjecture
- Twin prime conjecture
- infinitely many prime quadruplets
- infinitely many Mersenne primes (Lenstra-Pomerance-Wagstaff conjecture); equivalently, infinitely many even perfect numbers
- infinitely many regular primes, is their density $e^\{-^1!/\_2\}$
- infinitely many Cullen primes
- infinitely many palindromic primes in base 10
- infinitely many Fibonacci primes
- Is every Fermat number composite for $n\; >\; 4$?
- Is 78,557 the lowest Sierpinski number?
- Is 509,203 the lowest Riesel number?
- Fortune's conjecture (that no Fortunate number is composite)
- Polignac's conjecture
- Landau's problems

- abc conjecture
- existence of odd perfect numbers
- existence of quasiperfect numbers
- existence of odd weird numbers
- existence of Lychrel numbers
- Proof that 10 is a solitary number
- existence of Taxicab(5, 2, n) for n>1.
- Brocard's problem: existence of integers, n,m, such that n!+1=m
^{2}other than n=4,5,7

- Solving the Happy Ending problem for arbitrary $n$
- Finding matching upper and lower bounds for K-sets and halving lines
- The Hirsch conjecture on the lengths of shortest paths in the vertices and edges of a convex polytope

- The values of the Ramsey numbers, particularly $R(5,\; 5)$
- The values of the Van der Waerden numbers

- Hilbert's sixteenth problem
- Hadamard conjecture
- existence of perfect cuboids
- Existence of quadratic number fields being Euclidean but not norm-Euclidean

- 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?

- Erdős-Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
- The Hadwiger conjecture relating coloring to clique minors
- The Ringel-Kotzig conjecture on graceful labeling of trees
- The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
- Deriving a closed-form expression for the percolation threshold values, especially $p\_c$ (square site)
- Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow

- Schanuel's conjecture
- Lehmer's conjecture
- Pompeiu problem
- Is $gamma$ (the Euler-Mascheroni constant) irrational?

- Is every finitely presented periodic group finite?
- The inverse Galois problem
- For which positive integers m, n is the free Burnside group finite? In particular, is finite?

- Generalized star height problem
- Invariant subspace problem
- Modeling black hole mergers
- Problems in Latin squares
- Problems in loop theory and quasigroup theory

- Road coloring conjecture (Avraham Trahtman, 2007)
- The Angel problem (Various independent proofs, 2006)
- Stanley-Wilf conjecture (Gabor Tardos and Adam Marcus, 2004)
- Green–Tao theorem (Terence Tao, 2004)
- Poincaré conjecture (Solution by Grigori Perelman in 2002 now confirmed)
- Catalan's conjecture (Preda Mihăilescu, 2002)
- Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, 2001)
- The Langlands program for function fields (Laurent Lafforgue, 1999)
- Taniyama-Shimura conjecture (Wiles, Breuil, Conrad, Diamond, and Taylor, 1999)
- Kepler conjecture (Thomas Hales, 1998)
- Milnor conjecture (Vladimir Voevodsky, 1996)
- Fermat's Last Theorem (Andrew Wiles, 1994)
- Bieberbach conjecture (Louis de Branges, 1985)
- Four color theorem (Appel and Haken, 1977)

- Unsolved Problems in Number Theory, Logic and Cryptography
- Clay Institute Millennium Prize
- Unsolved problems page at MathWorld
- Winkelmann, Jörg, " Some Mathematical Problems". 9 March 2006.
- List of links to unsolved problems in mathematics, prizes and research.

- 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.

- 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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Last updated on Saturday October 11, 2008 at 08:26:35 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 11, 2008 at 08:26:35 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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