List of unsolved problems in mathematics

Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention.

This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative. Although this list may never be comprehensive, the problems listed here vary widely in both difficulty and importance.

Lists of unsolved problems in mathematics

Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

List Number of
problems
Number unsolved
or incompletely solved
Proposed by Proposed
in
Hilbert's problems[1] 23 15 David Hilbert 1900
Landau's problems[2] 4 4 Edmund Landau 1912
Taniyama's problems[3] 36 - Yutaka Taniyama 1955
Thurston's 24 questions[4][5] 24 - William Thurston 1982
Smale's problems 18 14 Stephen Smale 1998
Millennium Prize Problems 7 6[6] Clay Mathematics Institute 2000
Simon problems 15 <12[7][8] Barry Simon 2000
Unsolved Problems on Mathematics for the 21st Century[9] 22 - Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges[10][11] 23 - DARPA 2007
 
The Riemann zeta function, subject of the celebrated and influential unsolved problem known as the Riemann hypothesis

Millennium Prize Problems

Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:[6]

The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003.[12] However, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is unsolved.[13]

Notebooks

Unsolved problems

Algebra

 
In the Bloch sphere representation of a qubit, a SIC-POVM forms a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.

Group theory

 
The free Burnside group   is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups   are finite remains open.

Representation theory

Analysis

 
The area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.

Transcendental numbers and diophantine approximation

Combinatorics

Dynamical systems

 
A detail of the Mandelbrot set. It is not known whether the Mandelbrot set is locally connected or not.

Games and puzzles

Combinatorial games

Games with imperfect information

Geometry

Algebraic geometry

Covering and packing

Differential geometry

Discrete geometry

 
In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.

Euclidean geometry

Graph theory

Algebraic graph theory

Games on graphs

Graph coloring and labeling

 
An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.

Graph drawing and embedding

Restriction of graph parameters

Subgraphs

Word-representation of graphs

Miscellaneous graph theory

Model theory and formal languages

  • The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in   is a simple algebraic group over an algebraically closed field.
  • Generalized star height problem: can all regular languages be expressed using generalized regular expressions with limited nesting depths of Kleene stars?
  • For which number fields does Hilbert's tenth problem hold?
  • Kueker's conjecture[135]
  • The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for  -saturated models of a countable theory.[136]
  • Shelah's categoricity conjecture for  : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[136]
  • Shelah's eventual categoricity conjecture: For every cardinal   there exists a cardinal   such that if an AEC K with LS(K)<=   is categorical in a cardinal above   then it is categorical in all cardinals above  .[136][137]
  • The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
  • The stable forking conjecture for simple theories[138]
  • Tarski's exponential function problem: is the theory of the real numbers with the exponential function decidable?
  • 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?[139]
  • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[140]
  • Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is either finite,  , or  .
  • Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality   does it have a model of cardinality continuum?[141]
  • Do the Henson graphs have the finite model property?
  • Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
  • Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
  • If the class of atomic models of a complete first order theory is categorical in the  , is it categorical in every cardinal?[142][143]
  • Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
  • Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?[144]
  • Is the theory of the field of Laurent series over   decidable? of the field of polynomials over  ?
  • Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[145]
  • Determine the structure of Keisler's order.[146][147]

Probability theory

Number theory

General

 
6 is a perfect number because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them is odd.

Additive number theory

Algebraic number theory

  • Characterize all algebraic number fields that have some power basis.

Computational number theory

Diophantine equations

Prime numbers

 
Goldbach's conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.

Set theory

Note: These conjectures are about models of Zermelo-Frankel set theory with choice, and may not be able to be expressed in models of other set theories such as the various constructive set theories or non-wellfounded set theory.

Topology

 
The unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.

Problems solved since 1995

 
Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.

Algebra

Analysis

Combinatorics

Dynamical systems

Game theory

Geometry

21st century

20th century

Graph theory

Group theory

Number theory

21st century

20th century

Ramsey theory

Theoretical computer science

Topology

Uncategorised

2010s

2000s

See also

Notes

  1. ^ An aperiodic monotile has been discovered and the formal proof is awaiting publication. A preprint of the proof is available.[72]
  2. ^ A disproof has been announced, with a preprint made available on arXiv.[159]

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