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List of unsolved problems in mathematics

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Title: List of unsolved problems in mathematics  
Author: World Heritage Encyclopedia
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Subject: Perfect number, Erdős–Hajnal conjecture, Erdős–Gyárfás conjecture, Cycle double cover, Collatz conjecture
Collection: Conjectures, Lists of Lists, Lists of Unsolved Problems, Mathematics-Related Lists, Unsolved Problems in Mathematics
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List of unsolved problems in mathematics

This article lists some unsolved problems in mathematics. See individual articles for details and sources.


  • Millennium Prize Problems 1
  • Other still-unsolved problems 2
    • Additive number theory 2.1
    • Algebra 2.2
    • Algebraic geometry 2.3
    • Algebraic number theory 2.4
    • Analysis 2.5
    • Combinatorics 2.6
    • Discrete geometry 2.7
    • Euclidean geometry 2.8
    • Dynamical system 2.9
    • Graph theory 2.10
    • Group theory 2.11
    • Model theory 2.12
    • Number theory (general) 2.13
    • Number theory (prime numbers) 2.14
    • Partial differential equations 2.15
    • Ramsey theory 2.16
    • Set theory 2.17
    • Other 2.18
  • Problems solved recently 3
  • See also 4
  • References 5
    • Books discussing unsolved problems 5.1
    • Books discussing recently solved problems 5.2
  • External links 6

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


Algebraic geometry

Algebraic number theory



Discrete geometry

Euclidean geometry

Dynamical systems

Graph theory

Group theory

Model theory

  • 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)

Number theory (prime numbers)

Partial differential equations

Ramsey theory

Set theory


Problems solved recently

See also


  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,  .
  10. ^ Matschke, Benjamin (2014), "A survey on the square peg problem",  .
  11. ^  .
  12. ^ Wagner, Neal R. (1976), "The Sofa Problem", The American Mathematical Monthly 83 (3): 188–189,  
  13. ^  .
  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. ^  
  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.  
  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".  
  25. ^ Džamonja, Mirna, “Club guessing and the universal models.” On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
  26. ^  
  27. ^ Dobson, J. B. (June 2012) [2011], On Lerch's formula for the Fermat quotient, p. 15,  
  28. ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society 125: 1503–1509,  .
  29. ^ Franciscos Santos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics (Princeton University and Institute for Advanced Study) 176 (1): 383–412.  
  30. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (I)", Inventiones Mathematicae 178 (3): 485–504,  .
  31. ^  .

Books discussing unsolved problems

  • Fan Chung; Ron Graham (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters.  
  • Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry. Springer.  
  • Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer.  
  • Marcus Du Sautoy (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins.  
  • John Derbyshire (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press.  
  • Keith Devlin (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble.  
  • Vincent D. Blondel, Alexandre Megrestski (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press.  

Books discussing recently solved problems

  • Donal O'Shea (2007). The Poincaré Conjecture. Penguin.  
  • George G. Szpiro (2003). Kepler's Conjecture. Wiley.  
  • Mark Ronan (2006). Symmetry and the Monster. Oxford.  

External links

  • Unsolved Problems in Number Theory, Logic and Cryptography
  • Clay Institute Millennium Prize
  • List of links to unsolved problems in mathematics, prizes and research.
  • Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site
  • AIM Problem Lists
  • Unsolved Problem of the Week Archive. MathPro Press.
  • The Open Problems Project (TOPP), discrete and computational geometry problems
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