List of unsolved problems in mathematics

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

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

• Goldbach's conjecture and its weak version
• The values of g(k) and G(k) in Waring's problem
• Collatz conjecture (3n + 1 conjecture)
• Diophantine quintuples
• Gilbreath's conjecture
• Erdős conjecture on arithmetic progressions
• Erdős–Turán conjecture on additive bases
• Pollock octahedral numbers conjecture

Algebra

• Hilbert's sixteenth problem
• Hadamard conjecture
• Existence of perfect cuboids

Algebraic geometry

• André–Oort conjecture
• Bass conjecture
• Deligne conjecture
• Fröberg conjecture
• Fujita conjecture
• Hartshorne conjectures
• Jacobian conjecture
• Nakai conjecture
• n! 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 $\gamma$ (the Euler–Mascheroni constant), π + e, π − e, πe, 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, or transcendental? What is the irrationality measure of each of these numbers?
• 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 $S_n$
• 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 $k+1$ 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?
• 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 $n$
• Finding matching upper and lower bounds for K-sets and halving lines
• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2ⁿ smaller copies

Dynamical system

• Furstenberg conjecture – Is every invariant and ergodic measure for the $\times 2,\times 3$ 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
• 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 $p_c$ (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?

Number theory (general)

• abc 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=m² 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?

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 $e^{-1/2}$?
• 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 p²)?
• Are there infinitely many Wilson primes?
• Are there any Wall–Sun–Sun primes?
• Is every Fermat number 2²ⁿ + 1 composite for $n > 4$?
• Are all Fermat numbers square-free?
• Is 78,557 the lowest Sierpinski 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?

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 $R(5, 5)$
• 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^ℵ_ω < 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 ${\mathfrak p < \mathfrak t}$? (This problem was solved in a 2012 preprint by Malliaris and Shelah, who showed that ${\mathfrak p = \mathfrak t}$ is a theorem of ZFC.)
• Does the Generalized Continuum Hypothesis entail ${\diamondsuit(E^{\lambda^+}_{cf(\lambda)}})$ for every singular cardinal $\lambda$?

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)

See also: List of conjectures

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)
• 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)

See also

• Hilbert's 23 problems
• Smale's problems
• Timeline of mathematics
• List of conjectures#Open_problems
• List of statements undecidable in ZFC
• Lists of unsolved problems in mathematics

References

• Weisstein, Eric W., "Unsolved problems", MathWorld.
• Winkelmann, Jörg, "Some Mathematical Problems". 9 March 2006.
• Waldschmidt, Michael (2004). "Open Diophantine Problems". Moscow Mathematical Journal 4 (1): 245–305. ISSN 1609-3321. Zbl 1066.11030. 

Books discussing unsolved problems

• Fan Chung; Ron Graham (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 1-56881-111-X. 
• Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3. 
• Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer. ISBN 0-387-20860-7. 
• Victor Klee; Stan Wagon (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0-88385-315-9. 
• Marcus Du Sautoy (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0-06-093558-8. 
• John Derbyshire (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0-309-08549-7. 
• Keith Devlin (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN [[Special:BookSources/0-7607-8659-8|0-7607-8659-8[[Category:Articles with invalid ISBNs]]]] . 
• Vincent D. Blondel, Alexandre Megrestski (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0-691-11748-9. 

Books discussing recently solved problems

• Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1-84115-791-0. 
• Donal O'Shea (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9. 
• George G. Szpiro (2003). Kepler's Conjecture. Wiley. ISBN 0-471-08601-0. 
• Mark Ronan (2006). Symmetry and the Monster. Oxford. ISBN 0-19-280722-6. 

External links

• Unsolved Problems in Number Theory, Logic and Cryptography
• Clay Institute Millennium Prize
• List of links to unsolved problems in mathematics, prizes and research.
• Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site
• AIM Problem Lists
• Unsolved Problem of the Week Archive. MathPro Press.
• The Open Problems Project (TOPP), discrete and computational geometry problems