List of unsolved problems in mathematics

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This article reiterates the Millennium Prize list of unsolved problems in mathematics as of October 2014, and lists further unsolved problems in algebra, additive and algebraic number theories, analysis, combinatorics, algebraic, discrete, and Euclidean geometries, dynamical systems, partial differential equations, and graph, group, model, number, set and Ramsey theories, as well as miscellaneous unsolved problems. A list of problems solved since 1975 also appears, alongside some sources, general and specific, for the stated problems.

Millennium Prize Problems[edit]

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved, as of October 2014, :[citation needed]

The seventh problem, the Poincaré conjecture, has been solved.[citation needed] The smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere have two or more inequivalent smooth structures—is still unsolved.[citation needed]

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.[9]
  • Determine the structure of Keisler's order[10][11]
  • 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?[12]
  • The Stable Forking Conjecture for simple theories[13]
  • 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?[14]
  • Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[15]
  • If the class of atomic models of a complete first order theory is categorical in the \aleph_n, is it categorical in every cardinal?[16][17]
  • Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
  • Kueker's conjecture[18]
  • 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?[19]
  • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[20]

Number theory (general)[edit]

Number theory (prime numbers)[edit]

Partial differential equations[edit]

Ramsey theory[edit]

Set theory[edit]


Problems solved recently[when?][edit]


  1. ^ For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi ([1]), e ([2]), Khinchin's Constant ([3]), irrational numbers ([4]), transcendental numbers ([5]), and irrationality measures ([6]) at Wolfram MathWorld, all articles accessed 15 December 2014.
  2. ^ Michel Waldschmidt, 2008, "An introduction to irrationality and transcendence methods," at The University of Arizona The Southwest Center for Arithmetic Geometry 2008 Arizona Winter School, March 15-19, 2008 (Special Functions and Transcendence), see [7], accessed 15 December 2014.
  3. ^ John Albert, posting date unknown, "Some unsolved problems in number theory" [from Victor Klee & Stan Wagon, "Old and New Unsolved Problems in Plane Geometry and Number Theory"], in University of Oklahoma Math 4513 course materials, see [8], accessed 15 December 2014.
  4. ^ 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 .
  5. ^ Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society 61 (4): 346–253, doi:10.1090/noti1100 .
  6. ^ 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 .
  7. ^ Wagner, Neal R. (1976), "The Sofa Problem", The American Mathematical Monthly 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022 
  8. ^ 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 .
  9. ^ Shelah S, Classification Theory, North-Holland, 1990
  10. ^ Keisler, HJ, “Ultraproducts which are not saturated.” J. Symb Logic 32 (1967) 23—46.
  11. ^ Malliaris M, Shelah S, "A dividing line in simple unstable theories."
  12. ^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
  13. ^ Peretz, Assaf, “Geometry of forking in simple theories.” J. Symbolic Logic Volume 71, Issue 1 (2006), 347–359.
  14. ^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae 159 (1): 1–50. arXiv:math/9802134. 
  15. ^ Makowsky J, “Compactness, embeddings and definability,” in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
  16. ^ Baldwin, John T. (July 24, 2009). Categoricity. American Mathematical Society. ISBN 978-0821848937. Retrieved February 20, 2014. 
  17. ^ Shelah, Saharon. "Introduction to classification theory for abstract elementary classes". 
  18. ^ Hrushovski, Ehud, “Kueker's conjecture for stable theories.” Journal of Symbolic Logic Vol. 54, No. 1 (Mar., 1989), pp. 207–220.
  19. ^ 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. 
  20. ^ Džamonja, Mirna, “Club guessing and the universal models.” On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
  21. ^ 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. 
  22. ^ Dobson, J. B. (June 2012) [2011], On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907 
  23. ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society 125: 1503–1509, JSTOR 2162098 .
  24. ^ 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. [non-primary source needed]
  25. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (I)", Inventiones Mathematicae 178 (3): 485–504, doi:10.1007/s00222-009-0205-7 [non-primary source needed]
  26. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (II)", Inventiones Mathematicae 178 (3): 505–586, doi:10.1007/s00222-009-0206-6 [non-primary source needed]
  27. ^ 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 .[non-primary source needed]

Further reading[edit]

Books discussing unsolved problems[dated info][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[dated info][edit]

Other works[edit]

External links[edit]