Millennium Prize ProblemsOf 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.
 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)
- Gilbreath's conjecture
- Erdős conjecture on arithmetic progressions
- Erdős–Turán conjecture on additive bases
- Pollock octahedral numbers conjecture
 Number theory: prime numbers
- Catalan's Mersenne conjecture
- Twin prime conjecture
- 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 there infinitely many Wilson primes?
- Are there any Wall–Sun–Sun primes?
- Is every Fermat number 22n + 1 composite for n > 4?
- 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?
 General number theory
- abc conjecture
- Do any odd perfect numbers exist?
- 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
 Algebraic number theory
 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
 Ramsey theory
 General algebra
- Number of Magic squares (sequence A006052)
- Finding a formula for the probability that two elements chosen at random generate the symmetric group Sn
- 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 more than 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
 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 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 pc (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?
- the Jacobian conjecture
- Schanuel's conjecture
- Lehmer's conjecture
- Pompeiu problem
- Is γ (the Euler–Mascheroni constant) irrational?
- the Khabibullin’s conjecture on integral inequalities
- Fürstenberg 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
 Partial differential equations
 Group theory
- Is every finitely presented periodic group finite?
- The inverse Galois problem
- For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
 Set theory
- The problem of finding the ultimate core model, one that contains all large cardinals.
- If ℵω is a strong limit cardinal, then 2ℵω < ℵω1 (see Singular cardinals hypothesis). The best bound, ℵω4, 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 ℵω?
- Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
- Is it consistent that ?
- Does the Generalized Continuum Hypothesis entail for every singular cardinal λ?
- Invariant subspace problem
- Problems in Latin squares
- Problems in loop theory and quasigroup theory
- Dixmier conjecture
 Problems solved recently
- Circular law (Terence Tao and Van H. Vu, 2010)
- Hirsch conjecture (as announced by Francisco Santos Leal, 2010)
- 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, conjectured by Paul)
- 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
- Unsolved Problems in Number Theory, Logic and Cryptography
- Clay Institute Millennium Prize
- Weisstein, Eric W., "Unsolved problems" from MathWorld.
- Winkelmann, Jörg, "Some Mathematical Problems". 9 March 2006.
- List of links to unsolved problems in mathematics, prizes and research.
- Michael Waldschmidt (2004). "Open Diophantine Problems". Moscow Mathematical Journal 4 (1): 245–305.
- 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
 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 0060935588.
- John Derbyshire (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0309085497.
- Keith Devlin (2006). The Millennium Problems - The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 0-7607-8659-8.
- 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 1841157910.
- Donal O'Shea (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-846-14012-9.
- 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.