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* Number of ]s {{OEIS|id=A006052}} * Number of ]s {{OEIS|id=A006052}}
* Finding a formula for the probability that two elements chosen at random generate the ] <math>S_n</math> * Finding a formula for the probability that two elements chosen at random generate the ] <math>S_n</math>
* Frankl's ]: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets * Frankl's ]: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets

Revision as of 00:48, 25 June 2013

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (May 2008) (Learn how and when to remove this message)

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

Additive number theory

Algebra

Algebraic geometry

Algebraic number theory

Analysis

Combinatorics

Discrete geometry

  • Solving the Happy Ending problem for arbitrary n {\displaystyle 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 2 smaller copies

Dynamical system

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

Graph theory

Group theory

Number theory (general)

Number theory (prime numbers)

Partial differential equations

Ramsey theory

Set theory

Other

See also: List of conjectures

Problems solved recently

See also

References

  1. Weisstein, Eric W. "Pi". MathWorld.
  2. Weisstein, Eric W. "e". MathWorld.
  3. Weisstein, Eric W. "Khinchin's Constant". MathWorld.
  4. Weisstein, Eric W. "Irrational Number". MathWorld.
  5. Weisstein, Eric W. "Transcendental Number". MathWorld.
  6. Weisstein, Eric W. "Irrationality Measure". MathWorld.
  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. Dobson, J. B. (2012) , On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907 {{citation}}: Unknown parameter |month= ignored (help)
  11. Malliaris, M.; Shelah, S. (2012), Cofinality spectrum theorems in model theory, set theory and general topology, arXiv:1208.5424
  12. Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society, 125: 1503–1509 {{citation}}: Unknown parameter |Article Stable URL= ignored (help).
  13. Franciscos Santos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics. 176 (1). Princeton University and Institute for Advanced Study: 383–412. doi:10.4007/annals.2012.176.1.7.
  14. 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.
  15. 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • 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 ]. {{cite book}}: Check |isbn= value: invalid character (help)
  • Vincent D. Blondel, Alexandre Megrestski (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0-691-11748-9.

Books discussing recently solved problems

External links

Well-known unsolved problems by discipline
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