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.
The
Kourovka Notebook (
Russian: Коуровская тетрадь) is a collection of unsolved problems in
group theory, first published in 1965 and updated many times since.[14]
The
Sverdlovsk Notebook (
Russian: Свердловская тетрадь) is a collection of unsolved problems in
semigroup theory, first published in 1969 and updated many times since.[15][16][17]
Pierce–Birkhoff conjecture: every piecewise-polynomial is the maximum of a finite set of minimums of finite collections of polynomials.
Rota's basis conjecture: for matroids of rank with disjoint bases , it is possible to create an matrix whose rows are and whose columns are also bases.
Burnside problem: for which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems[22]
Herzog–Schönheim conjecture: if a finite system of left
cosets of subgroups of a group form a partition of , then the finite indices of said subgroups cannot be distinct.
The
inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
The
Brennan conjecture: estimating the integral of powers of the moduli of the derivative of
conformal maps into the open unit disk, on certain subsets of
The
Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
The
lonely runner conjecture – if runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance from each other runner) at some time?[31]
Map folding – various problems in map folding and stamp folding.
No-three-in-line problem – how many points can be placed in the grid so that no three of them lie on a line?
The
sunflower conjecture – can the number of size sets required for the existence of a sunflower of sets be bounded by an exponential function in for every fixed ?
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[33]
Does every positive integer generate a
juggler sequence terminating at 1?
Lyapunov function: Lyapunov's second method for stability – For what classes of
ODEs, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?
Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also
Hales-Jewett theorem and
nd game)[43]
Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a
boundedn-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?[47]
Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[50]
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.
Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.[53]
The
filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length[54]
The
Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds[55]
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[67]
Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?[68]
Danzer's problem and Conway's dead fly problem – do
Danzer sets of bounded density or bounded separation exist?[69]
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[76]
The
Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.[117]
Characterise
word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs[128])
Classify graphs with representation number 3, that is, graphs that can be
represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter[129]
The
second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?[132]
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]
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]
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]
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]
Casas-Alvero conjecture: if a polynomial of degree defined over a
field of
characteristic has a factor in common with its first through -th derivative, then must be the -th power of a linear polynomial?
n conjecture: a generalization of the abc conjecture to more than three integers.
abc conjecture: for any , is true for only finitely many positive such that .
Szpiro's conjecture: for any , there is some constant such that, for any elliptic curve defined over with minimal discriminant and conductor , we have .
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
Bunyakovsky conjecture: if an integer-coefficient polynomial has a positive leading coefficient, is irreducible over the integers, and has no common factors over all where is a positive integer, then is prime infinitely often.
Dickson's conjecture: for a finite set of linear forms with each , there are infinitely many for which all forms are
prime, unless there is some
congruence condition preventing it.
Dubner's conjecture: every even number greater than is the sum of two
primes which both have a
twin.
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?
New Mersenne conjecture: for any odd
natural number, if any two of the three conditions or , is prime, and is prime are true, then the third condition is also true.
Schinzel's hypothesis H that for every finite collection of nonconstant
irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers for which are all
primes, or there is some fixed divisor which, for all , divides some .
For any given integer a > 0, are there infinitely many
Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[156]
For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?
For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many
repunit primes to base b?
For any given integers , with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form with integer n ≥ 1?
McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)[169][170]
Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (
Jeff Cheeger, Aaron Naber, 2014)[263]
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^Morris, Walter D.; Soltan, Valeriu (2000), "The Erdős-Szekeres problem on points in convex position—a survey", Bull. Amer. Math. Soc., 37 (4): 437–458,
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^Brass, Peter; Moser, William; Pach, János (2005), "5.1 The Maximum Number of Unit Distances in the Plane", Research problems in discrete geometry, Springer, New York, pp. 183–190,
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^Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (May 28, 2023). "A chiral aperiodic monotile".
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^Arutyunyants, G.; Iosevich, A. (2004), "Falconer conjecture, spherical averages and discrete analogs", in
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^Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly, 59 (9): 606–611,
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^Pleanmani, Nopparat (2019), "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph", Discrete Mathematics, Algorithms and Applications, 11 (6): 1950068, 7,
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^Bousquet, Nicolas; Bartier, Valentin (2019), "Linear Transformations Between Colorings in Chordal Graphs", in Bender, Michael A.; Svensson, Ola; Herman, Grzegorz (eds.), 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, LIPIcs, vol. 144, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 24:1–24:15,
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^Pach, János;
Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, vol. 152,
American Mathematical Society, pp. 126–127.
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^Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7,
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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.
The
Kourovka Notebook (
Russian: Коуровская тетрадь) is a collection of unsolved problems in
group theory, first published in 1965 and updated many times since.[14]
The
Sverdlovsk Notebook (
Russian: Свердловская тетрадь) is a collection of unsolved problems in
semigroup theory, first published in 1969 and updated many times since.[15][16][17]
Pierce–Birkhoff conjecture: every piecewise-polynomial is the maximum of a finite set of minimums of finite collections of polynomials.
Rota's basis conjecture: for matroids of rank with disjoint bases , it is possible to create an matrix whose rows are and whose columns are also bases.
Burnside problem: for which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems[22]
Herzog–Schönheim conjecture: if a finite system of left
cosets of subgroups of a group form a partition of , then the finite indices of said subgroups cannot be distinct.
The
inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
The
Brennan conjecture: estimating the integral of powers of the moduli of the derivative of
conformal maps into the open unit disk, on certain subsets of
The
Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
The
lonely runner conjecture – if runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance from each other runner) at some time?[31]
Map folding – various problems in map folding and stamp folding.
No-three-in-line problem – how many points can be placed in the grid so that no three of them lie on a line?
The
sunflower conjecture – can the number of size sets required for the existence of a sunflower of sets be bounded by an exponential function in for every fixed ?
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[33]
Does every positive integer generate a
juggler sequence terminating at 1?
Lyapunov function: Lyapunov's second method for stability – For what classes of
ODEs, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?
Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also
Hales-Jewett theorem and
nd game)[43]
Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a
boundedn-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?[47]
Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[50]
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.
Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.[53]
The
filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length[54]
The
Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds[55]
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[67]
Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?[68]
Danzer's problem and Conway's dead fly problem – do
Danzer sets of bounded density or bounded separation exist?[69]
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[76]
The
Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.[117]
Characterise
word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs[128])
Classify graphs with representation number 3, that is, graphs that can be
represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter[129]
The
second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?[132]
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]
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]
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]
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]
Casas-Alvero conjecture: if a polynomial of degree defined over a
field of
characteristic has a factor in common with its first through -th derivative, then must be the -th power of a linear polynomial?
n conjecture: a generalization of the abc conjecture to more than three integers.
abc conjecture: for any , is true for only finitely many positive such that .
Szpiro's conjecture: for any , there is some constant such that, for any elliptic curve defined over with minimal discriminant and conductor , we have .
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
Bunyakovsky conjecture: if an integer-coefficient polynomial has a positive leading coefficient, is irreducible over the integers, and has no common factors over all where is a positive integer, then is prime infinitely often.
Dickson's conjecture: for a finite set of linear forms with each , there are infinitely many for which all forms are
prime, unless there is some
congruence condition preventing it.
Dubner's conjecture: every even number greater than is the sum of two
primes which both have a
twin.
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?
New Mersenne conjecture: for any odd
natural number, if any two of the three conditions or , is prime, and is prime are true, then the third condition is also true.
Schinzel's hypothesis H that for every finite collection of nonconstant
irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers for which are all
primes, or there is some fixed divisor which, for all , divides some .
For any given integer a > 0, are there infinitely many
Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[156]
For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?
For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many
repunit primes to base b?
For any given integers , with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form with integer n ≥ 1?
McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)[169][170]
Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (
Jeff Cheeger, Aaron Naber, 2014)[263]
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