List of unsolved problems in mathematics
Millennium Prize Problems
- 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)
- 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 ?
- 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 ap − 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 ?
- 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?
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
- Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell’s theorem)
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.
- 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 2n smaller copies
- 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 more than 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
- 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 (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
- Are (the Euler–Mascheroni constant), π+e, π–e, πe, π/e, πe, π√2, ππ, eπ2, ln π, 2e, ee, Catalan’s constant or Khinchin’s 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
- 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?
- 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 ?
- Invariant subspace problem
- Problems in Latin squares
- Problems in loop theory and quasigroup theory
- Dixmier conjecture
- Star height problem
- Assorted sphere packing problems, e.g. the densest irregular hypersphere packings
- Closed curve problem: Find (explicit) necessary an sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed. 
Problems solved recently
- Gromov’s problem on distortion of knots (John Pardon, 2011)
- Circular law (Terence Tao and Van H. Vu, 2010)
- Hirsch conjecture (as announced by 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, 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)
- Hilbert’s 23 problems
- Smale’s problems
- Timeline of mathematics
- List of conjectures#Open_problems
- List of statements undecidable in ZFC