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List of unsolved problems in mathematicsSince the Renaissance, every century has seen the solution of more mathematical problems than the century before, and yet many mathematical problems, both major and minor, still remain unsolved.Unsolved problems remain in multiple domains, including physics, computer science, algebra, additive and algebraic number theories, analysis, combinatorics, algebraic, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and miscellaneous unsolved problems. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.Lists of unsolved problems in mathematicsOver the course of time, several lists of unsolved mathematical problems have appeared.ListNumber of problemsProposed byProposed inHilbert's problems23David Hilbert1900Landau's problems4Edmund Landau1912Taniyama's problems[4]36Yutaka Taniyama1955Thurston's 24 questions[5][6]24William Thurston1982Smale's problems18Stephen Smale1998Millennium Prize problems7Clay Mathematics Institute2000Simon problems15Barry Simon2000Unsolved Problems on Mathematics for the 21st Century22Jair Minoro Abe, Shotaro Tanaka2001DARPA's math challenges[8][9]23DARPA2007Millennium Prize ProblemsOf the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved, as of 2018:P versus NPHodge conjectureRiemann hypothesisYang–Mills existence and mass gapNavier–Stokes existence and smoothnessBirch and Swinnerton-Dyer conjectureThe seventh problem, the Poincaré conjecture, has been solved.The smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.Unsolved problemsAlgebraHomological conjectures in commutative algebraHilbert's sixteenth problemHilbert's fifteenth problemHadamard conjectureJacobson's conjectureExistence of perfect cuboids and associated cuboid conjecturesZauner's conjecture: existence of SIC-POVMs in all dimensionsWild Problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problemsKöthe conjectureBirch–Tate conjectureSerre's conjecture IIBombieri–Lang conjectureFarrell–Jones conjectureBost conjectureUniformity conjectureKaplansky's conjectureKummer–Vandiver conjectureSerre's multiplicity conjecturesPierce–Birkhoff conjectureEilenberg–Ganea conjectureGreen's conjectureGrothendieck–Katz p-curvature conjectureSendov's conjectureAlgebraic geometryAbundance conjectureBass conjectureDeligne conjectureFröberg conjectureFujita conjectureHartshorne conjecturesThe Jacobian conjectureManin conjectureNakai conjectureResolution of singularities in characteristic pStandard conjectures on algebraic cyclesSection conjectureTate conjectureTermination of flipsVirasoro conjectureZariski multiplicity conjectureAnalysisSchanuel's conjecture and four exponentials conjectureLehmer's conjecturePompeiu problemAre [math] γ {\displaystyle \gamma } [/math] (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?[13][14][15]Khabibullin's conjecture on integral inequalitiesHilbert's thirteenth problemVitushkin's conjectureCombinatoricsFrankl'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[16]The lonely runner conjecture: if [math] k + 1 {\displaystyle k+1} [/math] runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance [math] 1 / ( k + 1 ) {\displaystyle 1/(k+1)} [/math] from each other runner) at some time?[17]Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?[18]Finding a function to model n-step self-avoiding walks.[19]The 1/3–2/3 conjecture: does every finite partially ordered set that is not totally ordered 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?[20]The uniqueness conjecture for Markov numbers[21]Give a combinatorial interpretation of the Kronecker coefficients.[22]Differential geometryFilling area conjectureHopf conjectureYau's conjectureDiscrete geometrySolving the happy ending problem for arbitrary [math] n {\displaystyle n} [/math][23]Finding matching upper and lower bounds for k-sets and halving lines[24]The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies[25]The Kobon triangle problem on triangles in line arrangements[26]The McMullen problem on projectively transforming sets of points into convex position[27]Ulam's packing conjecture about the identity of the worst-packing convex solid[28]Kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[29]How many unit distances can be determined by a set of n points in the Euclidean plane?[30]Euclidean geometryThe einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?[31]Falconer's conjecture that sets of Hausdorff dimension greater than [math] [/math]must have a distance set of nonzero Lebesgue measure[32]Inscribed square problem – does every Jordan curve have an inscribed square?[33]The Kakeya conjecture – double dimensional sets that contain a unit line segment in every dimension necessarily have Hausdorff dimension and Minkowski dimension equal toMoser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?[35]The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?[36]Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net?[37]The Thomson problem – what is the minimum energy configuration of N particles bound to the surface of a unit sphere that repel each other with a 1/r potential (or any potential in general)?g-conjectureCircle packing in an equilateral triangleCircle packing in an isosceles right triangleIs the Weaire–Phelan structure an optimal solution to the Kelvin problem?Lebesgue's universal covering problem – what is the convex shape in the plane of least area which provides an isometric cover for any shape of diameter one?Bellman's lost in a forest problem – for a given shape of forest find the shortest escape path which will intersect the edge of the forest at some point for any given starting point and direction inside the forest.Find the complete set of uniform 5-polytopes[38]Covering problem of RadoThe strong bellows conjecture – must the Dehn invariant of a self-intersection free flexible polyhedron stay constant as it flexes?Dissection into orthoschemes – is it possible for simplices of every dimension?[39]Dynamical systemsCollatz conjecture (3n + 1 conjecture)Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?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 groupsMLC conjecture – Is the Mandelbrot set locally connected?Weinstein conjecture – Does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theoryEremenko's conjecture that every component of the escaping set of an entire transcendental function is unboundedIs every reversible cellular automaton in three or more dimensions locally reversible?[40]Many problems concerning an outer billiard, for example show that outer billiards relative to almost every convex polygon has unbounded orbits.Games and puzzlesSudoku: What is the maximum number of givens for a minimal puzzle?[41] How many puzzles have exactly one solution?[41] How many minimal puzzles have exactly one solution?[41]Tic-tac-toe variants: Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy?[42]Graph theoryPaths and cycles in graphsBarnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle[43]Chvátal's toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian[44]The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice[45]The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs[46]The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree[47]The Lovász conjecture on Hamiltonian paths in symmetric graphs[48]The Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.[49]Graph coloring and labelingThe Erdős–Faber–Lovász conjecture on coloring unions of cliques[50]The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree[51]The Hadwiger conjecture relating coloring to clique minors[52]The Hadwiger–Nelson problem on the chromatic number of unit distance graphs[53]Hedetniemi's conjecture on the chromatic number of tensor products of graphs[54]Jaeger's Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph[55]The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index[56]The Ringel–Kotzig conjecture on graceful labeling of trees[57]The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree[58]Graph drawingThe Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number[59]The Blankenship–Oporowski conjecture on the book thickness of subdivisions[60]Conway's thrackle conjecture[61]Harborth's conjecture that every planar graph can be drawn with integer edge lengths[62]Negami's conjecture on projective-plane embeddings of graphs with planar covers[63]The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding[64]Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?[65]Universal point sets of subquadratic size for planar graphs[66]Miscellaneous graph theoryThe Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph[67]The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs[68]Jørgensen's conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph[69]Does a Moore graph with girth 5 and degree 57 exist?What is the largest possible pathwidth of an n-vertex cubic graph?The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flowVizing's conjecture on the domination number of cartesian products of graphs[72]Group theoryIs 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?Andrews–Curtis conjectureHerzog–Schönheim conjectureDoes generalized moonshine exist?Are there an infinite number of Leinster Groups?Number theoryGeneralGrand Riemann hypothesis Generalized Riemann hypothesis Riemann hypothesisn conjecture abc conjecture (Proof claimed in 2012, currently under review.)Hilbert's ninth problemHilbert's eleventh problemHilbert's twelfth problemCarmichael's totient function conjectureErdős–Straus conjecturePillai's conjectureHall's conjectureLindelöf hypothesisMontgomery's pair correlation conjectureHilbert–Pólya conjectureGrimm's conjectureLeopoldt's conjectureDo 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?Catalan–Dickson conjecture on aliquot sequencesDo 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, 7Beilinson conjectureLittlewood conjectureSzpiro's conjectureVojta's conjectureGoormaghtigh conjectureCongruent 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 amicable numbers which have opposite parity?Are there any pairs of relatively prime amicable numbers?Are there infinitely many betrothed numbers?Are there any pairs of betrothed numbers which have same parity?The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?Piltz divisor problem, especially Dirichlet's divisor problemExponent pair conjectureIs π a normal number (its digits are "random")?[86]Casas-Alvero conjectureSato–Tate conjectureFind value of De Bruijn–Newman constantWhich integers can be written as the sum of three perfect cubes?[87]Erdős–Moser problem: is 11 + 21 = 31 the only solution to the Erdős–Moser equation?Is there a covering system with odd distinct moduli?[88]Additive number theorySee also: Problems involving arithmetic progressionsBeal's conjectureFermat–Catalan conjectureGoldbach's conjectureThe values of g(k) and G(k) in Waring's problemLander, Parkin, and Selfridge conjectureGilbreath's conjectureErdős conjecture on arithmetic progressionsErdős–Turán conjecture on additive basesPollock octahedral numbers conjectureSkolem problemDetermine growth rate of rk(N) (see Szemerédi's theorem)Minimum overlap problemDo the Ulam numbers have a positive density?Algebraic number theoryAre there infinitely many real quadratic number fields with unique factorization (Class number problem)?Characterize all algebraic number fields that have some power basis.Stark conjectures (including Brumer–Stark conjecture)Kummer–Vandiver conjectureCombinatorial number theorySingmaster's conjecture: Is there a finite upper bound on the number of times that a number other than 1 can appear in Pascal's triangle?Computational number theoryInteger factorization: Can integer factorization be done in polynomial time?Prime numbersvtePrime number conjecturesBrocard's ConjectureCatalan's Mersenne conjectureAgoh–Giuga conjectureThe 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?New Mersenne conjectureErdős–Mollin–Walsh conjectureAre there infinitely many prime quadruplets?Are there infinitely many cousin primes?Are there infinitely many sexy primes?Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?Are there infinitely many Wagstaff primes?Are there infinitely many Sophie Germain primes?Are there infinitely many Pierpont primes?Are there infinitely many regular primes, and if so is their relative densityFor any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b?Are there infinitely many Cullen primes?Are there infinitely many Woodall primes?Are there infinitely many Carol primes?Are there infinitely many Kynea primes?Are there infinitely many palindromic primes to every base?Are there infinitely many Fibonacci primes?Are there infinitely many Lucas primes?Are there infinitely many Pell primes?Are there infinitely many Newman–Shanks–Williams primes?Are all Mersenne numbers of prime index square-free?Are there infinitely many Wieferich primes?Are there any Wieferich primes in base 47?Are there any composite c satisfying 2c − 1 ≡ 1 (mod c2)?For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[89]Can a prime p satisfy 2p − 1 ≡ 1 (mod p2) and 3p − 1 ≡ 1 (mod p2) simultaneously?[90]Are there infinitely many Wilson primes?Are there infinitely many Wolstenholme primes?Are there any Wall–Sun–Sun primes?For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)Is every Fermat number 22n + 1 composite for [math] n > 4 {\displaystyle n>4} [/math]?Are all Fermat numbers square-free?For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?Artin's conjecture on primitive rootsIs 78,557 the lowest Sierpiński number (so-called Selfridge's conjecture)?Is 509,203 the lowest Riesel number?Fortune's conjecture (that no Fortunate number is composite)Landau's problemsFeit–Thompson conjectureDoes every prime number appear in the Euclid–Mullin sequence?Does the converse of Wolstenholme's theorem hold for all natural numbers?Elliott–Halberstam conjectureProblems associated to Linnik's theoremFind the smallest Skewes' numberPartial differential equationsRegularity of solutions of Vlasov–Maxwell equationsRegularity of solutions of Euler equationsRamsey theoryThe values of the Ramsey numbers, particularly [math] R ( 5 , 5 ) {\displaystyle R(5,5)} [/math]The values of the Van der Waerden numbersSet theoryThe 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 Jónsson algebra on ℵω?Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?TopologyBorel conjectureHilbert–Smith conjectureNovikov conjectureUnknotting problemWhitehead conjectureZeeman conjectureOtherSee also: List of conjecturesList of unsolved problems in statisticsList of unsolved problems in computer scienceList of unsolved problems in physicsProblems in loop theory and quasigroup theoryProblems in Latin squaresInvariant subspace problemKaplansky's conjectures on group ringsPainlevé conjectureDixmier conjectureBaum–Connes conjectureProve Turing completeness for all unique elementary cellular automataGeneralized star height problemAssorted sphere packing problems, e.g. the densest irregular hypersphere packingsClosed curve problem: Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.[91]Keating–Snaith conjectureKung–Traub conjectureAtiyah conjecture on configurationsToeplitz' conjecture (open since 1911)Carathéodory conjectureWeight-monodromy conjectureBerry–Tabor conjectureBirkhoff conjectureGuralnick–Thompson conjectureMNOP conjectureMazur's conjecturesRendezvous problemScholz conjectureNirenberg–Treves conjectureQuantum unique ergodicity conjectureDensity hypothesisZhou conjectureErdős–Ulam problemProblems solved since 1995Pentagonal tiling (Michaël Rao, 2017)[92]Erdős–Burr conjecture (Choongbum Lee, 2017)[93]Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)[94]Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (A. Abdollahi, M. Zallaghi, 2015)[95]Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)[96]Erdős discrepancy problem (Terence Tao, 2015)[97]Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)[98]Anderson conjecture (Cheeger, Naber, 2014)[99]Gaussian correlation inequality (Thomas Royen, 2014)[100]Goldbach's weak conjecture (Harald Helfgott, 2013)[101][102][103]Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)[104][105] (and the Feichtinger's conjecture,Virtual Haken conjecture (Agol, Groves, Manning, 2012)[106] (and by work of Wise also virtually fibered conjecture)Hsiang–Lawson's conjecture (Brendle, 2012)[107]Willmore conjecture (Fernando Codá Marques and André Neves, 2012)[108]Ehrenpreis conjecture (Kahn, Markovic, 2011)[109]Hanna Neumann conjecture (Mineyev, 2011)[110]Bloch–Kato conjecture (Voevodsky, 2011)[111] (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture[112][113][114])Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)[115]Density theorem (Namazi, Souto, 2010)[116]Hirsch conjecture (Francisco Santos Leal, 2010)[117][118]Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)[119]Atiyah conjecture (Austin, 2009)[120]Kauffman–Harary conjecture (Matmann, Solis, 2009)[121]Surface subgroup conjecture (Kahn, Markovic, 2009)[122]Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)[123]Cobordism hypothesis (Jacob Lurie, 2008)[124]Full classification of finite simple groups (Harada, Solomon, 2008)Geometrization conjecture (proof was completed by Morgan and Tian in 2008[125] and it is based mostly on work of Grigori Perelman, 2002)[126]Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)[127][128][129]Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)[130]Erdős–Menger conjecture (Aharoni, Berger 2007)[131]Road coloring conjecture (Avraham Trahtman, 2007)[132]The angel problem (Various independent proofs, 2006)[133][134][135][136]Lax conjecture (Lewis, Parrilo, Ramana, 2005)[137]The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)[138]Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)[139]Robertson–Seymour theorem (Robertson, Seymour, 2004)[140]Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)[141] (and also Alon–Friedgut conjecture)Green–Tao theorem (Ben J. Green and Terence Tao, 2004)[142]Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)[143]Carpenter's rule problem (Connelly, Demaine, Rote, 2003)[144]Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)[145][146]Milnor conjecture (Vladimir Voevodsky, 2003)[147]Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)[148]Nagata's conjecture (Shestakov, Umirbaev, 2003)[149]Kirillov's conjecture (Baruch, 2003)[150]Poincaré conjecture (Grigori Perelman, 2002)[126]Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)[151]Kouchnirenko’s conjecture (Haas, 2002)[152]Vaught conjecture (Knight, 2002)[153]Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)[154]Catalan's conjecture (Preda Mihăilescu, 2002)[155]n! conjecture (Haiman, 2001)[156] (and also Macdonald positivity conjecture)Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)[157]Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)[158]Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)[159]Erdős–Stewart conjecture (Florian Luca, 2001)[160]Berry–Robbins problem (Atiyah, 2000)[161]Erdős–Graham problem (Croot, 2000)[162]Honeycomb conjecture (Thomas Hales, 1999)[163]Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)[164]Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)[165][166]Lafforgue's theorem (Laurent Lafforgue, 1998)[167]Kepler conjecture (Ferguson, Hales, 1998)[168]Dodecahedral conjecture (Hales, McLaughlin, 1998)[169]Ganea conjecture (Iwase, 1997)[170]Torsion conjecture (Merel, 1996)[171]Harary's conjecture (Chen, 1996)[172]Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)[173][174]

The biggest challenge in mathematical research are -Millennium Prize ProblemsOf the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved, as of 2019:[7]P versus NPHodge conjectureRiemann hypothesisYang–Mills existence and mass gapNavier–Stokes existence and smoothnessBirch and Swinnerton-Dyer conjectureThe seventh problem, the Poincaré conjecture, has been solved.[13]The smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.[14]Unsolved problemsAlgebraIn the Bloch sphere representation of a qubit, a SIC-POVM forms a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.Homological conjectures in commutative algebraFinite lattice representation problemHilbert's sixteenth problemHilbert's fifteenth problemHadamard conjectureJacobson's conjectureCrouzeix's conjectureExistence of perfect cuboids and associated cuboid conjecturesZauner's conjecture: existence of SIC-POVMs in all dimensionsWild Problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problemsKöthe conjectureBirch–Tate conjectureSerre's conjecture IIBombieri–Lang conjectureFarrell–Jones conjectureBost conjectureRota's basis conjectureUniformity conjectureKaplansky's conjecturesKummer–Vandiver conjectureSerre's multiplicity conjecturesPierce–Birkhoff conjectureEilenberg–Ganea conjectureGreen's conjectureGrothendieck–Katz p-curvature conjectureSendov's conjectureZariski–Lipman conjectureAlgebraic geometryAbundance conjectureBass conjectureDeligne conjectureDixmier conjectureFröberg conjectureFujita conjectureHartshorne conjectures[15]The Jacobian conjectureManin conjectureMaulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory[16]Nakai conjectureResolution of singularities in characteristicStandard conjectures on algebraic cyclesSection conjectureTate conjectureTermination of flipsVirasoro conjectureWeight-monodromy conjectureZariski multiplicity conjecture[17]AnalysisThe area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.The four exponentials conjecture on the transcendence of at least one of four exponentials of combinations of irrationals[18]Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials[19]The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy[20]Schanuel's conjecture on the transcendence degree of exponentials of linearly independent irrationals[18]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 rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?[21][22][23]Vitushkin's conjectureInvariant subspace problemKung–Traub conjecture[24]CombinatoricsFrankl'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[25]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?[26]Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?[27]Finding a function to model n-step self-avoiding walks.[28]The 1/3–2/3 conjecture: does every finite partially ordered set that is not totally orderedcontain 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?[29]Give a combinatorial interpretation of the Kronecker coefficients.[30]Open questions concerning Latin squaresDifferential geometryThe filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length[31]The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds[32]The spherical Bernstein's problem, a possible generalization of the original Bernstein's problemCarathéodory conjectureChern's conjecture (affine geometry)Chern's conjecture for hypersurfaces in spheresYau's conjectureYau's conjecture on the first eigenvalueClosed curve problem: Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.[33]Discrete geometryIn three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.Solving the happy ending problem for arbitrary [34]Finding matching upper and lower bounds for k-sets and halving lines[35]The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies[36]The Kobon triangle problem on triangles in line arrangements[37]The McMullen problem on projectively transforming sets of points into convex position[38]Tripod packing[39]Ulam's packing conjecture about the identity of the worst-packing convex solid[40]Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensionsWhat is the asymptotic growth rate of wasted space for packing unit squares into a half-integer square?[41]Kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[42]How many unit distances can be determined by a set of n points in the Euclidean plane?[43]Euclidean geometryBellman's lost in a forest problem – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation[44]Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?[45]Dissection into orthoschemes – is it possible for simplices of every dimension?[46]The einstein problem – does there exist a two-dimensional shape that forms the prototilefor an aperiodic tiling, but not for any periodic tiling?[47]The Erdős–Oler conjecture that when is a triangular number, packing circles in an equilateral triangle requires a triangle of the same size as packing circles[48]Falconer's conjecture that sets of Hausdorff dimension greater than in must have a distance set of nonzero Lebesgue measure[49]Inscribed square problem – does every Jordan curve have an inscribed square?[50]The Kakeya conjecture – do -dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to ?[51]The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem[52]Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one[53]Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?[54]The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?[55]Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net?[56]The Thomson problem – what is the minimum energy configuration of mutually-repelling particles on a unit sphere?[57]Uniform 5-polytopes – find and classify the complete set of these shapes[58]Covering problem of Rado – if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?[59]Toeplitz' conjecture (open since 1911)Atiyah conjecture on configurationsDynamical systemsA detail of the Mandelbrot set. It is not known whether the Mandelbrot set is locally connected or not.Collatz conjecture (3n + 1 conjecture)Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?Furstenberg conjecture – Is every invariant and ergodicmeasure for the action on the circle either Lebesgue or atomic?Margulis conjecture – Measure classification for diagonalizable actions in higher-rank groupsMLC conjecture – Is the Mandelbrot set locally connected?Weinstein conjecture – Does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theoryEremenko's conjecture that every component of the escaping set of an entire transcendental function is unboundedIs every reversible cellular automaton in three or more dimensions locally reversible?[60]Birkhoff conjecture: if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?[61]Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.Quantum unique ergodicity conjecture[62]Berry–Tabor conjecturePainlevé conjectureGames and puzzlesCombinatorial gamesSudoku:What is the maximum number of givens for a minimal puzzle?[63]How many puzzles have exactly one solution?[63]How many minimal puzzles have exactly one solution?[63]Tic-tac-toe variants:Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy?[64]What is the Turing completeness status of all unique elementary cellular automata?Games with imperfect informationRendezvous problemGraph theoryPaths and cycles in graphsBarnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle[65]Chvátal's toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian[66]The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice[67]The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs[68]The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree[69]The Lovász conjecture on Hamiltonian paths in symmetric graphs[70]The Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.[71]Graph coloring and labelingAn instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.The Erdős–Faber–Lovász conjecture on coloring unions of cliques[72]The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree[73]The Hadwiger conjecture relating coloring to clique minors[74]The Hadwiger–Nelson problem on the chromatic number of unit distance graphs[75]Jaeger's Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph[76]The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index[77]The Ringel–Kotzig conjecture on graceful labeling of trees[78]The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree[79]Graph drawingThe Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number[80]The Blankenship–Oporowski conjecture on the book thickness of subdivisions[81]Conway's thrackle conjecture[82]Harborth's conjecture that every planar graph can be drawn with integer edge lengths[83]Negami's conjecture on projective-plane embeddings of graphs with planar covers[84]The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding[85]Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?[86]Universal point sets of subquadratic size for planar graphs[87]Miscellaneous graph theoryConway's 99-graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?[88]The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph[89]The GNRS conjecture on whether minor-closed graph families have embeddings with bounded distortion[90]The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs[91]Jørgensen's conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph[92]Meyniel's conjecture that cop number is [93]Does a Moore graph with girth 5 and degree 57 exist?[94]What is the largest possible pathwidth of an n-vertex cubic graph?[95]The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.[96][97]The second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?[98]Sumner's conjecture: does every -vertex tournament contain as a subgraph every -vertex oriented tree?[99]Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow[100]Vizing's conjecture on the domination number of cartesian products of graphs[101]Group theoryThe free Burnside groupis finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groupsare finite remains open.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?Andrews–Curtis conjectureHerzog–Schönheim conjectureDoes generalized moonshine exist?Are there an infinite number of Leinster groups?Guralnick–Thompson conjecture[102]Problems in loop theory and quasigroup theory consider generalizations of groupsModel theory and formal languagesVaught's conjectureThe 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.[103]Determine the structure of Keisler's order[104][105]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?[106]The Stable Forking Conjecture for simple theories[107]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?[108]Shelah's eventual Categority conjecture: For every cardinal there exists a cardinal such that If an AEC K with LS(K)<= is categorical in a cardinal above then it is categorical in all cardinals above .[103][109]Shelah's categoricity conjecture for : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[103]Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[110]If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?[111][112]Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)Kueker's conjecture[113]Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?Do the Henson graphs have the finite model property?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?[114]The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[115]Generalized star height problemNumber theoryGeneral6 is a perfect number because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them are odd.Grand Riemann hypothesisGeneralized Riemann hypothesisRiemann hypothesisn conjectureabc conjectureHilbert's ninth problemHilbert's eleventh problemHilbert's twelfth problemCarmichael's totient function conjectureErdős–Straus conjectureErdős–Ulam problemPillai's conjectureHall's conjectureLindelöf hypothesis and its consequence, the density hypothesis for zeroes of the Riemann zeta function (see Bombieri–Vinogradov theorem)Montgomery's pair correlation conjectureHilbert–Pólya conjectureGrimm's conjectureLeopoldt's conjectureScholz conjectureDo 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?Catalan–Dickson conjecture on aliquot sequencesDo 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, 7Beilinson conjectureLittlewood conjectureSzpiro's conjectureVojta's conjectureGoormaghtigh conjectureCongruent 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 amicable numbers which have opposite parity?Are there any pairs of relatively prime amicable numbers?Are there infinitely many betrothed numbers?Are there any pairs of betrothed numbers which have same parity?The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?Piltz divisor problem, especially Dirichlet's divisor problemExponent pair conjectureIs π a normal number (its digits are "random")?[116]Casas-Alvero conjectureSato–Tate conjectureFind value of De Bruijn–Newman constantWhich integers can be written as the sum of three perfect cubes?[117]Erdős–Moser problem: is 11 + 21 = 31 the only solution to the Erdős–Moser equation?Is there a covering system with odd distinct moduli?[118]The uniqueness conjecture for Markov numbers[119]Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function[120]Additive number theoryBeal's conjectureFermat–Catalan conjectureGoldbach's conjectureThe values of g(k) and G(k) in Waring's problemLander, Parkin, and Selfridge conjectureGilbreath's conjectureErdős conjecture on arithmetic progressionsErdős–Turán conjecture on additive basesPollock octahedral numbers conjectureSkolem problemDetermine growth rate of rk(N) (see Szemerédi's theorem)Minimum overlap problemDo the Ulam numbers have a positive density?Algebraic number theoryAre there infinitely many real quadratic number fields with unique factorization (Class number problem)?Characterize all algebraic number fields that have some power basis.Stark conjectures (including Brumer–Stark conjecture)Kummer–Vandiver conjectureGreenberg's conjecturesComputational number theoryInteger factorization: Can integer factorization be done in polynomial time?Prime numbersGoldbach's conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.Brocard's ConjectureCatalan's Mersenne conjectureAgoh–Giuga conjectureThe 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?New Mersenne conjectureErdős–Mollin–Walsh conjectureAre there infinitely many prime quadruplets?Are there infinitely many cousin primes?Are there infinitely many sexy primes?Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?Are there infinitely many Wagstaff primes?Are there infinitely many Sophie Germain primes?Are there infinitely many Pierpont primes?Are there infinitely many regular primes, and if so is their relative density ?For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b?Are there infinitely many Cullen primes?Are there infinitely many Woodall primes?Are there infinitely many Carol primes?Are there infinitely many Kynea primes?Are there infinitely many palindromic primes to every base?Are there infinitely many Fibonacci primes?Are there infinitely many Lucas primes?Are there infinitely many Pell primes?Are there infinitely many Newman–Shanks–Williams primes?Are all Mersenne numbers of prime index square-free?Are there infinitely many Wieferich primes?Are there any Wieferich primes in base 47?Are there any composite c satisfying 2c − 1 ≡ 1 (mod c2)?For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[121]Can a prime p satisfy 2p − 1 ≡ 1 (mod p2) and 3p − 1 ≡ 1 (mod p2) simultaneously?[122]Are there infinitely many Wilson primes?Are there infinitely many Wolstenholme primes?Are there any Wall–Sun–Sun primes?For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)Is every Fermat number 22n + 1 composite for ?Are all Fermat numbers square-free?For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?Artin's conjecture on primitive rootsIs 78,557 the lowest Sierpiński number (so-called Selfridge's conjecture)?Is 509,203 the lowest Riesel number?Fortune's conjecture (that no Fortunate number is composite)Landau's problemsFeit–Thompson conjectureDoes every prime number appear in the Euclid–Mullin sequence?Does the converse of Wolstenholme's theorem hold for all natural numbers?Elliott–Halberstam conjectureProblems associated to Linnik's theoremFind the smallest Skewes' numberPartial differential equationsRegularity of solutions of Vlasov–Maxwell equationsRegularity of solutions of Euler equationsRamsey theoryThe values of the Ramsey numbers, particularlyThe values of the Van der Waerden numbersSet theoryThe 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 Jónsson algebra on ℵω?Without assuming the axiom of choice, can a nontrivial elementary embedding V→Vexist?Does the Generalized Continuum Hypothesis entail for every singular cardinal ?Does the Generalized Continuum Hypothesis imply the existence of an ℵ2-Suslin tree?Is OCA (Open coloring axiom) consistent with ?Assume ZF and that whenever there is a surjection from onto there is an injection from into . Does the Axiom of Choice hold?[123]TopologyThe unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.Baum–Connes conjectureBorel conjectureHilbert–Smith conjectureMazur's conjectures[124]Novikov conjectureUnknotting problemVolume conjectureWhitehead conjectureZeeman conjectureProblems solved since 1995Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.Hedetniemi's conjecture on the chromatic number of tensor products of graphs (2019)[125]Erdős sumset conjecture (2018)[126]McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel)[127][128]Pentagonal tiling (Michaël Rao, 2017)[129]Erdős–Burr conjecture (Choongbum Lee, 2017)[130]Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)[131][132]Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (A. Abdollahi, M. Zallaghi, 2015)[133]Main conjecture in Vinogradov's mean-value theorem(Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)[134]Erdős discrepancy problem (Terence Tao, 2015)[135]Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)[136]Anderson conjecture (Cheeger, Naber, 2014)[137]Gaussian correlation inequality (Thomas Royen, 2014)[138]Goldbach's weak conjecture (Harald Helfgott, 2013)[139][140][141]Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)[142][143] (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic and conjectures, Bourgain-Tzafriri conjecture and -conjecture)Virtual Haken conjecture (Agol, Groves, Manning, 2012)[144] (and by work of Wise also virtually fibered conjecture)Hsiang–Lawson's conjecture (Brendle, 2012)[145]Willmore conjecture (Fernando Codá Marques and André Neves, 2012)[146]Beck's 3-permutations conjecture (Newman, Nikolov, 2011)[147]Ehrenpreis conjecture (Kahn, Markovic, 2011)[148]Hanna Neumann conjecture (Mineyev, 2011)[149]Bloch–Kato conjecture (Voevodsky, 2011)[150] (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture[151][152][153])Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)[154]Density theorem (Namazi, Souto, 2010)[155]Hirsch conjecture (Francisco Santos Leal, 2010)[156][157]Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)[158]Atiyah conjecture (Austin, 2009)[159]Kauffman–Harary conjecture (Matmann, Solis, 2009)[160]Surface subgroup conjecture (Kahn, Markovic, 2009)[161]Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)[162]Cobordism hypothesis (Jacob Lurie, 2008)[163]Full classification of finite simple groups (Harada, Solomon, 2008)Geometrization conjecture, proven by Grigori Perelman[164] in a series of preprints in 2002-2003[165].Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)[166][167][168]Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)[169]Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Lu, 2007)[170]Erdős–Menger conjecture (Aharoni, Berger 2007)[171]Road coloring conjecture (Avraham Trahtman, 2007)[172]The angel problem (Various independent proofs, 2006)[173][174][175][176]Nirenberg–Treves conjecture (Nils Dencker, 2005)[177][178]Lax conjecture (Lewis, Parrilo, Ramana, 2005)[179]The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)[180]Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)[181]Robertson–Seymour theorem (Robertson, Seymour, 2004)[182]Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)[183] (and also Alon–Friedgut conjecture)Green–Tao theorem (Ben J. Green and Terence Tao, 2004)[184]Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)[185]Carpenter's rule problem (Connelly, Demaine, Rote, 2003)[186]Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)[187][188]Milnor conjecture (Vladimir Voevodsky, 2003)[189]Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)[190]Nagata's conjecture (Shestakov, Umirbaev, 2003)[191]Kirillov's conjecture (Baruch, 2003)[192]Poincaré conjecture (Grigori Perelman, 2002)[164]Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)[193]Kouchnirenko’s conjecture (Haas, 2002)[194]Vaught conjecture (Knight, 2002)[195]Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)[196]Catalan's conjecture (Preda Mihăilescu, 2002)[197]n! conjecture (Haiman, 2001)[198] (and also Macdonald positivity conjecture)Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)[199]Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)[200]Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)[201]Erdős–Stewart conjecture (Florian Luca, 2001)[202]Berry–Robbins problem (Atiyah, 2000)[203]Erdős–Graham problem (Croot, 2000)[204]Honeycomb conjecture (Thomas Hales, 1999)[205]Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)[206]Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)[207][208]Lafforgue's theorem (Laurent Lafforgue, 1998)[209]Kepler conjecture (Ferguson, Hales, 1998)[210]Dodecahedral conjecture (Hales, McLaughlin, 1998)[211]Ganea conjecture (Iwase, 1997)[212]Torsion conjecture (Merel, 1996)[213]Harary's conjecture (Chen, 1996)[214]Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)[215][2