# List of unsolved problems in mathematics

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved. These unsolved problems occur in multiple domains, including physics, computer science, algebra, analysis, combinatorics, algebraic, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and more. 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.

## Lists of unsolved problems in mathematics

Over the course of time, several lists of unsolved mathematical problems have appeared.

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 2019:

The seventh problem, the Poincaré conjecture, has been solved. 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

### 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

• Bellman's lost in a forest problem – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation
• Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?
• Dissection into orthoschemes – is it possible for simplices of every dimension?
• The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?
• 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
• Falconer's conjecture that sets of Hausdorff dimension greater than $d/2$  in $\mathbb {R} ^{d}$  must have a distance set of nonzero Lebesgue measure
• Inscribed square problem, also known as Toeplitz' conjecture – does every Jordan curve have an inscribed square?
• The Kakeya conjecture – do $n$ -dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to $n$ ?
• The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem
• Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one
• Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?
• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
• Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net?
• The Thomson problem – what is the minimum energy configuration of $n$  mutually-repelling particles on a unit sphere?
• Uniform 5-polytopes – find and classify the complete set of these shapes
• 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?
• Atiyah conjecture on configurations

### 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.

### Model theory and formal languages

• 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.
• Determine the structure of Keisler's order
• 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?
• The Stable Forking Conjecture for simple theories
• 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?
• 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 $\mu (\lambda )$ .
• 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.
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?
• 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.)
• Kueker's conjecture
• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
• Do the Henson graphs have the finite model property?
• 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?
• Generalized star height problem

### 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.

#### 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.

### Ramsey theory

• The values of the Ramsey numbers, particularly $R(5,5)$
• The values of the Van der Waerden numbers