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]

Euclidean geometry[edit]

Dynamical systems[edit]

Graph theory[edit]

Group theory[edit]

Model theory[edit]

  • Vaught's conjecture
  • The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in \aleph_0 is a simple algebraic group over an algebraically closed field.
  • The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for \aleph_1-saturated models of a countable theory.[14]
  • Determine the structure of Keisler's order[15][16]
  • The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
  • Is the theory of the field of Laurent series over \mathbb{Z}_p decidable? of the field of polynomials over \mathbb{C}?
  • (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[17]
  • The Stable Forking Conjecture for simple theories[18]
  • For which number fields does Hilbert's tenth problem hold?
  • Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality \aleph_{\omega_1} does it have a model of cardinality continuum?[19]
  • Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[20]
  • If the class of atomic models of a complete first order theory is categorical in the \aleph_n, is it categorical in every cardinal?[21][22]
  • Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
  • Kueker's conjecture[23]
  • Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
  • Lachlan's decision problem
  • Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
  • Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
  • The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?[24]
  • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[25]

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. ^ Socolar, Joshua E. S.; Taylor, Joan M. (2012), Forcing nonperiodicity with a single tile, The Mathematical Intelligencer 34 (1): 18–28, arXiv:1009.1419, doi:10.1007/s00283-011-9255-y, MR 2902144 .
  10. ^ Matschke, Benjamin (2014), A survey on the square peg problem, Notices of the American Mathematical Society 61 (4): 346–253, doi:10.1090/noti1100 .
  11. ^ Norwood, Rick; Poole, George; Laidacker, Michael (1992), The worm problem of Leo Moser, Discrete and Computational Geometry 7 (2): 153–162, doi:10.1007/BF02187832, MR 1139077 .
  12. ^ Wagner, Neal R. (1976), The Sofa Problem, The American Mathematical Monthly 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022 
  13. ^ Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338 .
  14. ^ Shelah S, Classification Theory, North-Holland, 1990
  15. ^ Keisler, HJ, “Ultraproducts which are not saturated.” J. Symb Logic 32 (1967) 23—46.
  16. ^ Malliaris M, Shelah S, "A dividing line in simple unstable theories."
  17. ^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
  18. ^ Peretz, Assaf, “Geometry of forking in simple theories.” J. Symbolic Logic Volume 71, Issue 1 (2006), 347–359.
  19. ^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae 159 (1): 1–50. arXiv:math/9802134. 
  20. ^ Makowsky J, “Compactness, embeddings and definability,” in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
  21. ^ Baldwin, John T. (July 24, 2009). Categoricity. American Mathematical Society. ISBN 978-0821848937. Retrieved February 20, 2014. 
  22. ^ Shelah, Saharon. "Introduction to classification theory for abstract elementary classes". 
  23. ^ Hrushovski, Ehud, “Kueker's conjecture for stable theories.” Journal of Symbolic Logic Vol. 54, No. 1 (Mar., 1989), pp. 207–220.
  24. ^ Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B 97 (3): 293–333. arXiv:math/0512218. doi:10.1016/j.jctb.2006.05.008. 
  25. ^ Džamonja, Mirna, “Club guessing and the universal models.” On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
  26. ^ 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. 
  27. ^ Dobson, J. B. (June 2012) [2011], On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907 
  28. ^ Barros, Manuel (1997), General Helices and a Theorem of Lancret, American Mathematical Society 125: 1503–1509, JSTOR 2162098 .
  29. ^ 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. 
  30. ^ 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 .
  31. ^ 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]

  • 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 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 978-0-7607-8659-8. 
  • 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[edit]

External links[edit]