# List of unsolved problems in mathematics

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

This article is a composite of notable unsolved problems derived from many sources, including but not limited to lists considered authoritative. The list is not comprehensive, for at least the reason that entries may not be updated at the time of viewing. This list includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole.

## 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 unresolved
or incompletely resolved
Proposed by Proposed in
Hilbert's problems 23 15 David Hilbert 1900
Landau's problems 4 4 Edmund Landau 1912
Taniyama's problems 36 - Yutaka Taniyama 1955
Thurston's 24 questions 24 - William Thurston 1982
Smale's problems 18 14 Stephen Smale 1998
Millennium Prize problems 7 6 Clay Mathematics Institute 2000
Simon problems 15 <12 Barry Simon 2000
Unsolved Problems on Mathematics for the 21st Century 22 - Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges 23 - DARPA 2007

### Millennium Prize Problems

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved as of August, 2021:

The seventh problem, the Poincaré conjecture, has been solved; 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 still unsolved.

## Unsolved problems

### Algebra

#### Notebook problems

• The Dneister Notebook (Dnestrovskaya Tetrad) collects several hundred unresolved problems in algebra, particularly ring theory and modulus theory.
• The Erlagol Notebook (Erlagolskaya Tetrad) collects unresolved problems in algebra and model theory.

### Combinatorics

#### Conjectures and problems

• The 1/3–2/3 conjecture: does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?
• Problems in Latin squares - Open questions concerning Latin squares
• The lonely runner conjecture: if $k+1$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance $1/(k+1)$ from each other runner) at some time?
• No-three-in-line problem
• Frankl's union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets

#### Other

• The values of the Dedekind numbers $M(n)$ for $n\geq 9$ .
• Give a combinatorial interpretation of the Kronecker coefficients.
• The values of the Ramsey numbers, particularly $R(5,5)$ • Finding a function to model n-step self-avoiding walks.
• The values of the Van der Waerden numbers

### Geometry

#### Covering and packing

##### Conjectures and problems
• Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.
• The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?
• The Erdős–Oler conjecture that when $n$ is a triangular number, packing $n-1$ circles in an equilateral triangle requires a triangle of the same size as packing $n$ circles
• The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24
• Reinhardt's conjecture that the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets
• Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
• Square packing in a square: what is the asymptotic growth rate of wasted space?
• Ulam's packing conjecture about the identity of the worst-packing convex solid

#### 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.

### Graph theory

#### 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.

### Group theory The free Burnside group $B(2,3)$ is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups $B(m,n)$ are finite remains open.

#### Notebook problems

• The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.

### Model theory and formal languages

#### Conjectures and problems

• 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.
• Generalized star height problem
• For which number fields does Hilbert's tenth problem hold?
• Kueker's conjecture
• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for $\aleph _{1}$ -saturated models of a countable theory.
• Shelah's categoricity conjecture for $L_{\omega _{1},\omega }$ : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.
• Shelah's eventual categoricity conjecture: For every cardinal $\lambda$ there exists a cardinal $\mu (\lambda )$ such that if an AEC K with LS(K)<= $\lambda$ is categorical in a cardinal above $\mu (\lambda )$ then it is categorical in all cardinals above $\mu (\lambda )$ .
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• The Stable Forking Conjecture for simple theories
• Tarski's exponential function problem
• 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?
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?
• Vaught's conjecture

#### Open questions

• 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?
• 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 $\aleph _{n}$ , is it categorical in every cardinal?
• 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.)
• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?
• Is the theory of the field of Laurent series over $\mathbb {Z} _{p}$ decidable? of the field of polynomials over $\mathbb {C}$ ?
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?

#### Other

• Determine the structure of Keisler's order

### 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 are odd.

#### Algebraic number theory

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

#### 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.

## Problems solved since 1995

### Analysis

• Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013) (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic $KS_{r}$ and $KS'_{r}$ conjectures, Bourgain-Tzafriri conjecture and $R_{\epsilon }$ -conjecture)