Unsolved problems in mathematics
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This article lists some unsolved problems in mathematics. See individual articles for details and sources.
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[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
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.
[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
- 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
- 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
[edit] General number theory
- 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
- Solving the Happy Ending problem for arbitrary n
- existence of Taxicab(5, 2, n) for n>1.
[edit] Ramsey theory
- The values of the Ramsey numbers, particularly R(5,5)
- The values of the Van der Waerden numbers
[edit] General algebra
- Hilbert's sixteenth problem
- Hadamard conjecture
- existence of perfect cuboids
- Existence of quadratic number fields being Euclidean but not norm-Euclidean
[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 Hadwiger conjecture
- The Ringel-Kotzig conjecture
- 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
- Problems in Latin squares
- Problems in loop theory and quasigroup theory
- See also: List of conjectures
[edit] Problems solved recently
- Road coloring conjecture (Avraham Trahtman, 2007)
- The Angel problem (Various independent proofs, 2006)
- 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
- 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.
[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.
- 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.
[edit] 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.
[edit] Resources
- Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site
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