BREAKING NEWS
List of unsolved problems in mathematics

## Summary

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]

## 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.

#### Notebook problems

• The Dniester Notebook (Russian: Днестровская тетрадъ) lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory.[15]
• The Erlagol Notebook (Russian: Эрлаголъская тетрадъ) lists unsolved problems in algebra and model theory.[16]

### Analysis

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

### Combinatorics

• 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?[24]
• Problems in Latin squares – open questions concerning Latin squares
• The lonely runner conjecture – if ${\displaystyle k+1}$  runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance ${\displaystyle 1/(k+1)}$  from each other runner) at some time?[25]
• The sunflower conjecture: can the number of ${\displaystyle k}$  size sets required for the existence of a sunflower of ${\displaystyle r}$  sets be bounded by an exponential function in ${\displaystyle k}$  for every fixed ${\displaystyle r>2}$ ?
• No-three-in-line problem – how many points can be placed in the ${\displaystyle n\times n}$  grid so that no three of them lie on a line?
• 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[26]

### Dynamical systems

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

### Geometry

#### Algebraic geometry

• Are infinite sequences of flips possible in dimensions greater than 3?
• Resolution of singularities in characteristic ${\displaystyle p}$

#### Covering and packing

• 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?[40]
• The Erdős–Oler conjecture: when ${\displaystyle n}$  is a triangular number, packing ${\displaystyle n-1}$  circles in an equilateral triangle requires a triangle of the same size as packing ${\displaystyle n}$  circles[41]
• The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[42]
• Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[43]
• 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?[44]
• Ulam's packing conjecture about the identity of the worst-packing convex solid[45]

#### 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.
• Finding matching upper and lower bounds for k-sets and halving lines[56]
• Tripod packing:[57] how many tripods can have their apexes packed into a given cube?

### 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 ${\displaystyle B(2,3)}$  is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups ${\displaystyle B(m,n)}$  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.[125]

### Model theory and formal languages

• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ${\displaystyle \aleph _{0}}$  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[126]
• The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for ${\displaystyle \aleph _{1}}$ -saturated models of a countable theory.[127]
• Shelah's categoricity conjecture for ${\displaystyle L_{\omega _{1},\omega }}$ : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[127]
• Shelah's eventual categoricity conjecture: For every cardinal ${\displaystyle \lambda }$  there exists a cardinal ${\displaystyle \mu (\lambda )}$  such that if an AEC K with LS(K)<= ${\displaystyle \lambda }$  is categorical in a cardinal above ${\displaystyle \mu (\lambda )}$  then it is categorical in all cardinals above ${\displaystyle \mu (\lambda )}$ .[127][128]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• The stable forking conjecture for simple theories[129]
• 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?[130]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[131]
• Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is either finite, ${\displaystyle \aleph _{0}}$ , or ${\displaystyle 2^{\aleph _{0}}}$ .
• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ${\displaystyle \aleph _{\omega _{1}}}$  does it have a model of cardinality continuum?[132]
• 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 ${\displaystyle \aleph _{n}}$ , is it categorical in every cardinal?[133][134]
• 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?[135]
• Is the theory of the field of Laurent series over ${\displaystyle \mathbb {Z} _{p}}$  decidable? of the field of polynomials over ${\displaystyle \mathbb {C} }$ ?
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[136]
• Determine the structure of Keisler's order.[137][138]

### 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.
• n conjecture: a generalization of the abc conjecture to more than three integers.
• abc conjecture: for any ${\displaystyle \epsilon >0}$ , ${\displaystyle {\text{rad}}(abc)^{1+\epsilon }  is true for only finitely many positive ${\displaystyle a,b,c}$  such that ${\displaystyle a+b=c}$ .
• Szpiro's conjecture: for any ${\displaystyle \epsilon >0}$ , there is some constant ${\displaystyle C(\epsilon )}$  such that, for any elliptic curve ${\displaystyle E}$  defined over ${\displaystyle \mathbb {Q} }$  with minimal discriminant ${\displaystyle \Delta }$  and conductor ${\displaystyle f}$ , we have ${\displaystyle |\Delta |\leq C(\epsilon )\cdot f^{6+\epsilon }}$ .
• Hardy–Littlewood zeta-function conjectures
• Hilbert's eleventh problem: classify quadratic forms over algebraic number fields.
• Hilbert's ninth problem: find the most general reciprocity law for the norm residues of ${\displaystyle k}$ -th order in a general algebraic number field, where ${\displaystyle k}$  is a power of a prime.
• Hilbert's twelfth problem: extend the Kronecker–Weber theorem on Abelian extensions of ${\displaystyle \mathbb {Q} }$  to any base number field.
• Grand Riemann hypothesis: do the nontrivial zeros of all automorphic L-functions lie on the critical line ${\displaystyle 1/2+it}$  with real ${\displaystyle t}$ ?
• Generalized Riemann hypothesis: do the nontrivial zeros of all Dirichlet L-functions lie on the critical line ${\displaystyle 1/2+it}$  with real ${\displaystyle t}$ ?
• Riemann hypothesis: do the nontrivial zeros of the Riemann zeta function lie on the critical line ${\displaystyle 1/2+it}$  with real ${\displaystyle t}$ ?
• André–Oort conjecture: is every irreducible component of the Zariski closure of a set of special points in a Shimura variety a special subvariety?
• Beilinson's conjectures
• Brocard's problem: are there any integer solutions to ${\displaystyle n!+1=m^{2}}$  other than ${\displaystyle n=4,5,7}$ ?
• Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than ${\displaystyle 1}$ ?
• Casas-Alvero conjecture: if a polynomial of degree ${\displaystyle d}$  defined over a field ${\displaystyle K}$  of characteristic ${\displaystyle 0}$  has a factor in common with its first through ${\displaystyle d-1}$ -th derivative, then must ${\displaystyle f}$  be the ${\displaystyle d}$ -th power of a linear polynomial?
• Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating.
• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are congruent numbers.
• Erdős–Moser problem: is ${\displaystyle 1^{1}+2^{1}=3^{1}}$  the only solution to the Erdős–Moser equation?
• Erdős–Straus conjecture: for every ${\displaystyle n\geq 2}$ , there are positive integers ${\displaystyle x,y,z}$  such that ${\displaystyle 4/n=1/x+1/y+1/z}$ .
• Erdős–Ulam problem: is there a dense set of points in the plane all at rational distances from one-another?
• Exponent pair conjecture: for all ${\displaystyle \epsilon >0}$ , is the pair ${\displaystyle (\epsilon ,1/2+\epsilon )}$  an exponent pair?
• The Gauss circle problem: how far can the number of integer points in a circle centered at the origin be from the area of the circle?
• Goormaghtigh conjecture on solutions to ${\displaystyle (x^{m}-1)/(x-1)=(y^{n}-1)/(y-1)}$  where ${\displaystyle x>y>1}$  and ${\displaystyle m,n>2}$ .
• Grimm's conjecture: each element of a set of consecutive composite numbers can be assigned a distinct prime number that divides it.
• Hall's conjecture: for any ${\displaystyle \epsilon >0}$ , there is some constant ${\displaystyle c(\epsilon )}$  such that either ${\displaystyle y^{2}=x^{3}}$  or ${\displaystyle |y^{2}-x^{3}|>c(\epsilon )x^{1/2-\epsilon }}$ .
• Hilbert–Pólya conjecture: the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator.
• Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function[139]
• Lehmer's totient problem: if ${\displaystyle \phi (n)}$  divides ${\displaystyle n-1}$ , must ${\displaystyle n}$  be prime?
• Leopoldt's conjecture: a p-adic analogue of the regulator of an algebraic number field does not vanish.
• Lindelöf hypothesis that for all ${\displaystyle \epsilon >0}$ , ${\displaystyle \zeta (1/2+it)=o(t^{\epsilon })}$
• Littlewood conjecture: for any two real numbers ${\displaystyle \alpha ,\beta }$ , ${\displaystyle \liminf _{n\rightarrow \infty }n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0}$ , where ${\displaystyle \Vert x\Vert }$  is the distance from ${\displaystyle x}$  to the nearest integer.
• Mahler's 3/2 problem that no real number ${\displaystyle x}$  has the property that the fractional parts of ${\displaystyle x(3/2)^{n}}$  are less than ${\displaystyle 1/2}$  for all positive integers ${\displaystyle n}$ .
• Montgomery's pair correlation conjecture: the normalized pair correlation function between pairs of zeros of the Riemann zeta function is the same as the pair correlation function of random Hermitian matrices.
• Newman's conjecture: the partition function satisfies any arbitrary congruence infinitely often.
• Pillai's conjecture: for any ${\displaystyle A,B,C}$ , the equation ${\displaystyle Ax^{m}-By^{n}=C}$  has finitely many solutions when ${\displaystyle m,n}$  are not both ${\displaystyle 2}$ .
• Piltz divisor problem on bounding ${\displaystyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(log(x))}$
• Dirichlet's divisor problem: the specific case of the Piltz divisor problem for ${\displaystyle k=1}$
• Ramanujan–Petersson conjecture: a number of related conjectures that are generalizations of the original conjecture.
• Sato–Tate conjecture: also a number of related conjectures that are generalizations of the original conjecture.
• Scholz conjecture: the length of the shortest addition chain producing ${\displaystyle 2^{n}-1}$  is at most ${\displaystyle n-1}$  plus the length of the shortest addition chain producing ${\displaystyle n}$ .
• Do Siegel zeros exist?
• Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?[140]
• The uniqueness conjecture for Markov numbers[141] that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation.
• Vojta's conjecture on heights of points on algebraic varieties over algebraic number fields.

• Beal's conjecture: for all integral solutions to ${\displaystyle A^{x}+B^{y}=C^{z}}$  where ${\displaystyle x,y,z>2}$ , all three numbers ${\displaystyle A,B,C}$  must share some prime factor.
• Erdős conjecture on arithmetic progressions that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long arithmetic progressions.
• Erdős–Turán conjecture on additive bases: if ${\displaystyle B}$  is an additive basis of order ${\displaystyle 2}$ , then the number of ways that positive integers ${\displaystyle n}$  can be expressed as the sum of two numbers in ${\displaystyle B}$  must tend to infinity as ${\displaystyle n}$  tends to infinity.
• Fermat–Catalan conjecture: there are finitely many distinct solutions ${\displaystyle (a^{m},b^{n},c^{k})}$  to the equation ${\displaystyle a^{m}+b^{n}=c^{k}}$  with ${\displaystyle a,b,c}$  being positive coprime integers and ${\displaystyle m,n,k}$  being positive integers satisfying ${\displaystyle 1/m+1/n+1/k<1}$ .
• Gilbreath's conjecture on consecutive applications of the unsigned forward difference operator to the sequence of prime numbers.
• Goldbach's conjecture: every even natural number greater than ${\displaystyle 2}$  is the sum of two prime numbers.
• Lander, Parkin, and Selfridge conjecture: if the sum of ${\displaystyle m}$  ${\displaystyle k}$ -th powers of positive integers is equal to a different sum of ${\displaystyle n}$  ${\displaystyle k}$ -th powers of positive integers, then ${\displaystyle m+n\geq k}$ .
• Lemoine's conjecture: all odd integers greater than ${\displaystyle 5}$  can be represented as the sum of an odd prime number and an even semiprime.
• Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set ${\displaystyle \{1,\ldots ,2n\}}$
• Pollock's conjectures
• Skolem problem: can an algorithm determine if a constant-recursive sequence contains a zero?
• The values of g(k) and G(k) in Waring's problem

#### Algebraic number theory

• 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.
• Agoh–Giuga conjecture on the Bernoulli numbers that ${\displaystyle p}$  is prime if and only if ${\displaystyle pB_{p-1}\equiv -1{\pmod {p}}}$
• Artin's conjecture on primitive roots that if an integer is neither a perfect square nor ${\displaystyle -1}$ , then it is a primitive root modulo infinitely many prime numbers ${\displaystyle p}$
• Brocard's conjecture: there are always at least ${\displaystyle 4}$  prime numbers between consecutive squares of prime numbers, aside from ${\displaystyle 2^{2}}$  and ${\displaystyle 3^{2}}$ .
• Bunyakovsky conjecture: if an integer-coefficient polynomial ${\displaystyle f}$  has a positive leading coefficient, is irreducible over the integers, and has no common factors over all ${\displaystyle f(x)}$  where ${\displaystyle x}$  is a positive integer, then ${\displaystyle f(x)}$  is prime infinitely often.
• Catalan's Mersenne conjecture: some Catalan–Mersenne number is composite and thus all Catalan–Mersenne numbers are composite after some point.
• Dickson's conjecture: for a finite set of linear forms ${\displaystyle a_{1}+b_{1}n,\ldots ,a_{k}+b_{k}n}$  with each ${\displaystyle b_{i}\geq 1}$ , there are infinitely many ${\displaystyle n}$  for which all forms are prime, unless there is some congruence condition preventing it.
• Dubner's conjecture: every number greater than ${\displaystyle 2408}$  is the sum of two primes which both have twins.
• Elliott–Halberstam conjecture on the distribution of prime numbers in arithmetic progressions.
• Erdős–Mollin–Walsh conjecture: no three consecutive numbers are all powerful.
• Feit–Thompson conjecture: for all distinct prime numbers ${\displaystyle p}$  and ${\displaystyle q}$ , ${\displaystyle (p^{q}-1)/(p-1)}$  does not divide ${\displaystyle (q^{p}-1)/(q-1)}$
• Fortune's conjecture that no Fortunate number is composite.
• The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
• Gillies' conjecture on the distribution of prime divisors of Mersenne numbers.
• Goldbach conjecture: all even natural numbers greater than ${\displaystyle 2}$  are the sum of two prime numbers.
• Landau's problems
• Problems associated to Linnik's theorem
• New Mersenne conjecture: for any odd natural number ${\displaystyle p}$ , if any two of the three conditions ${\displaystyle p=2^{k}\pm 1}$  or ${\displaystyle p=4^{k}\pm 3}$ , ${\displaystyle 2^{p}-1}$  is prime, and ${\displaystyle (2^{p}+1)/3}$  is prime are true, then the third condition is true.
• Polignac's conjecture: for all positive even numbers ${\displaystyle n}$ , there are infinitely many prime gaps of size ${\displaystyle n}$ .
• Schinzel's hypothesis H that for every finite collection ${\displaystyle \{f_{1},\ldots ,f_{k}\}}$  of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers ${\displaystyle n}$  for which ${\displaystyle f_{1}(n),\ldots ,f_{k}(n)}$  are all primes, or there is some fixed divisor ${\displaystyle m>1}$  which, for all ${\displaystyle n}$ , divides some ${\displaystyle f_{i}(n)}$ .
• Selfridge's conjecture: is 78,557 the lowest Sierpiński number?
• Twin prime conjecture: there are infinitely many twin primes.
• Does the converse of Wolstenholme's theorem hold for all natural numbers?

### 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.

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