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 (3n + 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 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 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 \gamma (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 rational, algebraic irrational, or transcendental? 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 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? 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 MLC conjecture - Is the Mandelbrot set locally connected ? 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? Conway's thrackle conjecture 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 \aleph_0 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 \aleph_1-saturated models of a countable theory.[9] Determine the structure of Keisler's order[10][11] 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 \mathbb{Z}_p decidable? of the field of polynomials over \mathbb{C}? (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[12] The Stable Forking Conjecture for simple theories[13] 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 \aleph_{\omega_1} does it have a model of cardinality continuum?[14] Is there a logic satisfying the interpolation theorem which is compact?[15] If the class of atomic models of a complete first order theory is categorical in the \aleph_n, is it categorical in every cardinal?[16][17] Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure) Kueker's conjecture[18] 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?[19] The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[20] 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=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? 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 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 ap − 1 ≡ 1 (mod p2)?[21] Can a prime p satisfy 2p − 1 ≡ 1 (mod p2) and 3p − 1 ≡ 1 (mod p2) simultaneously?[22] Are there infinitely many Wilson primes? Are there infinitely many Wolstenholme 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 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 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 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 {\mathfrak p < \mathfrak t}? (This problem was solved in a 2012 preprint by Malliaris and Shelah,[23] 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.[24] 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[25]) Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008[26]) 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)[27] 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) |

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