Unsolved problems in mathematics
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This article lists some currently unsolved problems in mathematics. See individual article for details and sources.
Contents |
[edit] Millennium Prize Problems
Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six ones yet to be solved are:
- 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
[edit] Other still-unsolved problems
[edit] Additive number theory
- 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
[edit] Number theory: prime numbers
- 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
- infinitely many Cullen primes
- infinitely many palindromic primes in base 10
- Is every Fermat number composite for n > 4?
- Is 78,557 the lowest Sierpinski number?
- Is 509,203 the lowest Riesel number?
[edit] General number theory
- abc conjecture
- existence of odd perfect numbers
- existence of quasiperfect numbers
- existence of odd weird numbers
- Proof that 196 is the smallest Lychrel number
- Proof that 10 is a solitary number
- Solving the Happy Ending problem for arbitrary n
[edit] Ramsey theory
- The values of the Ramsey numbers, particularly R(5,5)
- The values of the Van der Warden numbers
[edit] General algebra
[edit] 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 Sn
[edit] Graph theory
- Erdős-Gyárfás conjecture
- The Graph isomorphism problem
- The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
- Deriving a closed-form expression for the percolation threshold values, especially pc (square site)
[edit] Analysis
- Schanuel's conjecture
- Lehmer's conjecture
- Pompeiu problem
- Is γ (the Euler-Mascheroni constant) irrational?
[edit] Group theory
- Is every finitely presented periodic group finite?
- The inverse Galois problem
[edit] Other
- Generalized star height problem
- Invariant subspace problem
- Modeling black hole mergers
- The Angel problem
See also the list of conjectures.
[edit] Problems solved recently
- Stanley-Wilf conjecture (Gabor Tardos and Adam Marcus, 2004)
- Poincaré conjecture (Solution by Grigori Perelman in 2002 now confirmed)
- Catalan's conjecture (Preda Mihăilescu, 2002)
- Kato's conjecture (Auscher, Hofmann, Lacey, 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)
[edit] See also
[edit] Quotes
[edit] References
- 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.
[edit] 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.
[edit] Books discussing recently solved problems
- Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1841157910.
