“At first, you should read this paragraph. I want to let you know what mathematics is. After that, I will show you a list of unsolved problems in mathematics. Are these too hard for you so as to solve? The answer depends on your thoughts. If you really love math and have spent a lot of time on doing math research. I think you can solve them someday, but not really everyone, even me. Addtionally, your achievement still depends on your luckiness”

**Mathematics** is the abstract study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.^{}^{
}

Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid’s *Elements*. Since the pioneering work ofGiuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosenaxioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.^{
}

Galileo Galilei (1564–1642) said, “The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth.” Carl Friedrich Gauss (1777–1855) referred to mathematics as “the Queen of the Sciences”.^{}Benjamin Peirce (1809–1880) called mathematics “the science that draws necessary conclusions”. David Hilbert said of mathematics: “We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.” Albert Einstein (1879–1955) stated that “as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” French mathematician Claire Voisin states “There is creative drive in mathematics, it’s all about movement trying to express itself.” ^{
}

Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

This article lists some **unsolved problems in mathematics**. See individual articles for details and sources.

## Contents

- 1 Millennium Prize Problems
- 2 Other still-unsolved problems
- 2.1 Additive number theory
- 2.2 Algebra
- 2.3 Algebraic geometry
- 2.4 Algebraic number theory
- 2.5 Analysis
- 2.6 Combinatorics
- 2.7 Discrete geometry
- 2.8 Dynamical system
- 2.9 Graph theory
- 2.10 Group theory
- 2.11 Model theory
- 2.12 Number theory (general)
- 2.13 Number theory (prime numbers)
- 2.14 Partial differential equations
- 2.15 Ramsey theory
- 2.16 Set theory
- 2.17 Other

- 3 Problems solved recently
- 4 See also
- 5 References

## Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved:

- P versus NP
- Hodge conjecture
- Riemann hypothesis
- Yang–Mills existence and mass gap
- Navier–Stokes existence and smoothness
- Birch and Swinnerton-Dyer conjecture.

The seventh problem, the Poincaré conjecture, has been solved. The smooth four-dimensional Poincaré conjecture is still unsolved. That is, can a four-dimensional topological sphere have two or more inequivalent smooth structures?

## Other still-unsolved problems

### Additive number theory

- Beal’s conjecture
- Goldbach’s conjecture (Proof claimed for weak version in 2013)
- The values of
*g*(*k*) and*G*(*k*) in Waring’s problem - Collatz conjecture (3
*n*+ 1 conjecture) - Lander, Parkin, and Selfridge conjecture
- Diophantine quintuples
- Gilbreath’s conjecture
- Erdős conjecture on arithmetic progressions
- Erdős–Turán conjecture on additive bases
- Pollock octahedral numbers conjecture

### Algebra

### Algebraic geometry

- André–Oort conjecture
- Bass conjecture
- Deligne conjecture
- Fröberg conjecture
- Fujita conjecture
- Hartshorne conjectures
- Jacobian conjecture
- Manin conjecture
- Nakai conjecture
- Resolution of singularities in characteristic p
- Standard conjectures on algebraic cycles
- Section conjecture
- Virasoro conjecture
- Witten conjecture
- Zariski multiplicity conjecture

### Algebraic number theory

- Are there infinitely many real quadratic number fields with unique factorization?
- Brumer–Stark conjecture
- Characterize all algebraic number fields that have some power basis.

### Analysis

- The Jacobian conjecture
- Schanuel’s conjecture
- Lehmer’s conjecture
- Pompeiu problem
- Are (the Euler–Mascheroni constant), π +
*e*, π −*e*, π*e*, π/*e*, π^{e}, π^{√2}, π^{π}, e^{π2}, ln π, 2^{e},*e*^{e}, Catalan’s constant or Khinchin’s constant rational, algebraic irrational, ortranscendental? What is the irrationality measure of each of these numbers?^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]} - The Khabibullin’s conjecture on integral inequalities

### Combinatorics

- Number of magic squares (sequence A006052 in OEIS)
- Finding a formula for the probability that two elements chosen at random generate the symmetric group
- 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
- 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?
- Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?
- The 1/3–2/3 conjecture: does every finite partially ordered set 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? - Conway’s thrackle conjecture

### Discrete geometry

- Solving the Happy Ending problem for arbitrary
- Finding matching upper and lower bounds for K-sets and halving lines
- The Hadwiger conjecture on covering
*n*-dimensional convex bodies with at most 2^{n}smaller copies

### Dynamical system

- Furstenberg conjecture – Is every invariant and ergodic measure for the action on the circle either Lebesgue or atomic?
- Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups

### Graph theory

- Barnette’s conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle
- The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs (proof claimed –
^{[9]}) - The Erdős–Hajnal conjecture on finding large homogeneous sets in graphs with a forbidden induced subgraph
- The Hadwiger conjecture relating coloring to clique minors
- The Erdős–Faber–Lovász conjecture on coloring unions of cliques
- The total coloring conjecture
- The list coloring conjecture
- The Ringel–Kotzig conjecture on graceful labeling of trees
- The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
- Deriving a closed-form expression for the percolation threshold values, especially (square site)
- Tutte’s conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow
- The Reconstruction conjecture and New digraph reconstruction conjecture concerning whether or not a graph is recognizable by the vertex deleted subgraphs.
- The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice.
- Does a Moore graph with girth 5 and degree 57 exist?

### Group theory

- Is every finitely presented periodic group finite?
- The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
- For which positive integers
*m*,*n*is the free Burnside group B(*m*,*n*) finite? In particular, is B(2, 5) finite? - Is every group surjunctive?

### Model theory

- Vaught’s conjecture
- The Cherlin-Zilber conjecture: A simple group whose first-order theory is stable in 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 -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 decidable? of the field of polynomials over ?
- (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 does it have a model of cardinality continuum?
^{ } - Is there a logic satisfying the interpolation theorem which is compact?
^{ } - If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?
^{}^{ } - Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
- Kueker’s conjecture
^{ } - Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
- Lachlan’s decision problem
- Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
- Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
- 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?
^{[20]} - The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?
^{[21]}

### Number theory (general)

*abc*conjecture (Proof claimed in 2012, currently under review.)- Erdős–Straus conjecture
- Do any odd perfect numbers exist?
- Are there infinitely many perfect numbers?
- Do quasiperfect numbers exist?
- Do any odd weird numbers exist?
- Do any Lychrel numbers exist?
- Is 10 a solitary number?
- Do any Taxicab(5, 2, n) exist for
*n*>1? - Brocard’s problem: existence of integers,
*n*,*m*, such that*n*!+1=*m*^{2}other than*n*=4,5,7 - Distribution and upper bound of mimic numbers
- Littlewood conjecture
- Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell’s theorem)
- Lehmer’s totient problem: if φ(
*n*) divides*n*− 1, must*n*be prime? - Are there infinitely many amicable numbers?
- Are there any pairs of relatively prime amicable numbers?

### Number theory (prime numbers)

- Catalan’s Mersenne conjecture
- Twin prime conjecture
- 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?
- Are there infinitely many prime quadruplets?
- Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
- Are there infinitely many Sophie Germain primes?
- Are there infinitely many regular primes, and if so is their relative density ?
- Are there infinitely many Cullen primes?
- Are there infinitely many palindromic primes in base 10?
- Are there infinitely many Fibonacci primes?
- Are all Mersenne numbers of prime index square-free?
- Are there infinitely many Wieferich primes?
- Are there for every a ≥ 2 infinitely many primes p such that a
^{p − 1}≡ 1 (mod*p*^{2})?^{[22]} - Can a prime
*p*satisfy 2^{p − 1}≡ 1 (mod*p*^{2}) and 3^{p − 1}≡ 1 (mod*p*^{2}) simultaneously?^{[23]} - Are there infinitely many Wilson primes?
- Are there infinitely many Wolstenholme primes?
- Are there any Wall–Sun–Sun primes?
- Is every Fermat number 2
^{2n}+ 1 composite for ? - Are all Fermat numbers square-free?
- Is 78,557 the lowest Sierpiński number?
- Is 509,203 the lowest Riesel number?
- Fortune’s conjecture (that no Fortunate number is composite)
- Polignac’s conjecture
- Landau’s problems
- Does every prime number appear in the Euclid–Mullin sequence?
- Does the converse of Wolstenholme’s theorem hold for all natural numbers?
- Elliott–Halberstam conjecture

### Partial differential equations

- Regularity of solutions of Vlasov–Maxwell equations
- Regularity of solutions of Euler equations

### Ramsey theory

- The values of the Ramsey numbers, particularly
- The values of the Van der Waerden numbers

### Set theory

- The problem of finding the ultimate core model, one that contains all large cardinals.
- If ℵ
_{ω}is a strong limit cardinal, then 2^{ℵω}< ℵ_{ω1}(see Singular cardinals hypothesis). The best bound, ℵ_{ω4}, was obtained by Shelah using his pcf theory. - Woodin’s Ω-hypothesis.
- Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
- (Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?
- Does there exist a Jonsson algebra on ℵ
_{ω}? - Without assuming the axiom of choice, can a nontrivial elementary embedding
*V*→*V*exist? - Is it consistent that ? (This problem was solved in a 2012 preprint by Malliaris and Shelah,
^{[24]}who showed that is a theorem of ZFC.) - Does the Generalized Continuum Hypothesis entail for every singular cardinal ?

### Other

- Invariant subspace problem
- Problems in Latin squares
- Problems in loop theory and quasigroup theory
- Dixmier conjecture
- Baum–Connes conjecture
- Generalized star height problem
- Assorted sphere packing problems, e.g. the densest irregular hypersphere packings
- 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.
^{ } - Toeplitz’ conjecture (open since 1911)

## Problems solved recently

- Gromov’s problem on distortion of knots (John Pardon, 2011)
- Circular law (Terence Tao and Van H. Vu, 2010)
- Hirsch conjecture (Francisco Santos Leal, 2010)
- Serre’s modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)
^{} - Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2007)
- Road coloring conjecture (Avraham Trahtman, 2007)
- The Angel problem (Various independent proofs, 2006)
- The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)
- Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)
- Green–Tao theorem (Ben J. Green and Terence Tao, 2004)
- Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003, conjectured by Paul Erdős)
^{ } - Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)
- Poincaré conjecture (Grigori Perelman, 2002)
- Catalan’s conjecture (Preda Mihăilescu, 2002)
- Kato’s conjecture (Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, 2001)
- The Langlands correspondence for function fields (Laurent Lafforgue, 1999)
- Taniyama–Shimura conjecture (Wiles, Breuil, Conrad, Diamond, and Taylor, 1999)
- Kepler conjecture (Thomas Hales, 1998)
- Milnor conjecture (Vladimir Voevodsky, 1996)
- Fermat’s Last Theorem (Andrew Wiles and Richard Taylor, 1995)
- Bieberbach conjecture (Louis de Branges, 1985)
- Princess and monster game (Shmuel Gal, 1979)
- Four color theorem (Appel and Haken, 1977)