List of unsolved problems in mathematics

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Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, and yet many mathematical problems, both major and minor, still elude solution.[1] Most graduate students, in order to earn a Ph.D. in mathematics, are expected to produce new, original mathematics. That is, they are expected to solve problems that are not routine, and which cannot be solved by standard methods.

In the context of this article, a mathematical problem is a statement (conjecture) that no one knows whether it is true or not. The problem is to determine whether it is true or false. The task of the problem solver is to either produce a proof of the statement or a proof that the statement is false.

An unsolved problem in mathematics does not refer to the kind of problem found as an exercise in a textbook, but rather to the answer to a major question or a general method that provides a solution to an entire class of problems. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems receive considerable attention. This article reiterates the list of Millennium Prize Problems of unsolved problems in mathematics (includes problems of physics and computer science) as of August 2015, 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.

Lists of unsolved problems in mathematics[edit]

Over the course of time, several lists of unsolved mathematical problems have appeared. The following is a listing of those lists.

List Number of problems Proposed by Proposed in
Hilbert's problems 23 David Hilbert 1900
Landau's problems 4 Edmund Landau 1912
Taniyama's problems[2] 36 Yutaka Taniyama 1955
Smale's problems 18 Stephen Smale 1998
Millennium Prize problems 7 Clay Mathematics Institute 2000
Unsolved Problems on Mathematics for the 21st Century[3] 22 Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges[4][5] 23 DARPA 2007

Millennium Prize Problems[edit]

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved, as of August 2015:[6]

The seventh problem, the Poincaré conjecture, has been solved.[7] The smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.[8]

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.[19]
  • Determine the structure of Keisler's order[20][21]
  • 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?[22]
  • The Stable Forking Conjecture for simple theories[23]
  • 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?[24]
  • Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[25]
  • If the class of atomic models of a complete first order theory is categorical in the \aleph_n, is it categorical in every cardinal?[26][27]
  • Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
  • Kueker's conjecture[28]
  • 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?[29]
  • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[30]

Number theory (general)[edit]

Number theory (prime numbers)[edit]

Partial differential equations[edit]

Ramsey theory[edit]

Set theory[edit]


Problems solved since 1975[edit]


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  39. ^ Santos, Franciscos (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. 
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  41. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (I)", Inventiones Mathematicae 178 (3): 485–504, doi:10.1007/s00222-009-0205-7 
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Further reading[edit]

Books discussing unsolved problems[dated info][edit]

  • Fan Chung; Graham, Ron (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 1-56881-111-X. 
  • Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3. 
  • Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 0-387-20860-7. 
  • Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0-88385-315-9. 
  • Du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0-06-093558-8. 
  • Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0-309-08549-7. 
  • Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8. 
  • Blondel, Vincent D.; Megrestski, Alexandre (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]