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[edit]

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[edit]

Additive number theory[edit]


Algebraic geometry[edit]

Algebraic number theory[edit]



Discrete geometry[edit]

Dynamical system[edit]

Graph theory[edit]

Group theory[edit]

Model theory[edit]

Number theory (general)[edit]

Number theory (prime numbers)[edit]

Partial differential equations[edit]

Ramsey theory[edit]

Set theory[edit]


Problems solved recently[edit]

See also[edit]


  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. ^ Heckman and Krakovski, “Erdös-Gyárfás Conjecture for Cubic Planar Graphs.” Elect. J. of Combinatorics 20 (2013),
  10. ^ Shelah S, Classification Theory, North-Holland, 1990
  11. ^ Keisler, HJ, “Ultraproducts which are not saturated.” J. Symb Logic 32 (1967) 23—46.
  12. ^ Malliaris M, Shelah S, "A dividing line in simple unstable theories."
  13. ^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479-506.
  14. ^ Peretz, Assaf, “Geometry of forking in simple theories.” J. Symbolic Logic Volume 71, Issue 1 (2006), 347-359.
  15. ^ Shelah S, “Borel sets with large squares.” Fund. Math. 159 (1999) 1—50. arxiv:9802134
  16. ^ Makowsky J, “Compactness, embeddings and definability,” in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645-715.
  17. ^ Baldwin, John, Categoricity in AECs, AMS 2009.
  18. ^ S Shelah, “Introduction to classification theory for abstract elementary classes.”
  19. ^ Hrushovski, Ehud, “Kueker's conjecture for stable theories.” Journal of Symbolic Logic Vol. 54, No. 1 (Mar., 1989), pp. 207-220.
  20. ^ Cherlin G, Shelah S, “Universal graphs with a forbidden subtree.” J. Comb Thy Ser B, vol 97 pps. 293—333. arxiv:0512218
  21. ^ D\v{z}amonja, Mirna, “Club guessing and the universal models.” On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
  22. ^ 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. 
  23. ^ Dobson, J. B. (06 2012) [2011], On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907 
  24. ^ Malliaris, M.; Shelah, S. (2012), Cofinality spectrum theorems in model theory, set theory and general topology, arXiv:1208.5424 
  25. ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society 125: 1503–1509  Unknown parameter |Article Stable URL= ignored (help).
  26. ^ 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. 
  27. ^ 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 .
  28. ^ 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[edit]

Books discussing recently solved problems[edit]

External links[edit]