List of unsolved problems in mathematics
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Since 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.^{[1]} Prizes are often awarded for the solution to a longstanding problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention. 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.
Contents
 1 Lists of unsolved problems in mathematics

2 Unsolved problems
 2.1 Algebra
 2.2 Algebraic geometry
 2.3 Analysis
 2.4 Combinatorics
 2.5 Differential geometry
 2.6 Discrete geometry
 2.7 Euclidean geometry
 2.8 Dynamical systems
 2.9 Games and puzzles
 2.10 Graph theory
 2.11 Group theory
 2.12 Model theory
 2.13 Number theory
 2.14 Partial differential equations
 2.15 Ramsey theory
 2.16 Set theory
 2.17 Other
 3 Problems solved since 1995
 4 References
 5 Further reading
 6 External links
Lists of unsolved problems in mathematics
Over the course of time, several lists of unsolved mathematical problems have appeared.
List  Number of problems  Proposed by  Proposed in 

Hilbert's problems^{[2]}  23  David Hilbert  1900 
Landau's problems^{[3]}  4  Edmund Landau  1912 
Taniyama's problems^{[4]}  36  Yutaka Taniyama  1955 
Thurston's 24 questions^{[5]}^{[6]}  24  William Thurston  1982 
Smale's problems  18  Stephen Smale  1998 
Millennium Prize problems  7  Clay Mathematics Institute  2000 
Unsolved Problems on Mathematics for the 21st Century^{[7]}  22  Jair Minoro Abe, Shotaro Tanaka  2001 
DARPA's math challenges^{[8]}^{[9]}  23  DARPA  2007 
Millennium Prize Problems
Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved, as of 2017:^{[10]}
 P versus NP
 Hodge conjecture
 Riemann hypothesis
 Yang–Mills existence and mass gap
 Navier–Stokes existence and smoothness
 Birch and SwinnertonDyer conjecture
The seventh problem, the Poincaré conjecture, has been solved.^{[11]} The smooth fourdimensional Poincaré conjecture—that is, whether a fourdimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.^{[12]}
Unsolved problems
Algebra
 Homological conjectures in commutative algebra
 Hilbert's sixteenth problem
 Hilbert's fifteenth problem
 Hadamard conjecture
 Jacobson's conjecture
 Existence of perfect cuboids and associated cuboid conjectures
 Zauner's conjecture: existence of SICPOVMs in all dimensions
 Wild Problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
 Köthe conjecture
 Birch–Tate conjecture
 Serre's conjecture II
 Bombieri–Lang conjecture
 Farrell–Jones conjecture
 Bost conjecture
 Uniformity conjecture
 Kaplansky's conjecture
 Kummer–Vandiver conjecture
 Serre's multiplicity conjectures
 Pierce–Birkhoff conjecture
 Eilenberg–Ganea conjecture
 Green's conjecture
 Grothendieck–Katz pcurvature conjecture
 Sendov's conjecture
Algebraic geometry
 Bass conjecture
 Deligne conjecture
 Fröberg conjecture
 Fujita conjecture
 Hartshorne conjectures
 The Jacobian conjecture
 Manin conjecture
 Nakai conjecture
 Resolution of singularities in characteristic p
 Standard conjectures on algebraic cycles
 Section conjecture
 Tate conjecture
 Virasoro conjecture
 Zariski multiplicity conjecture
Analysis
 Schanuel's conjecture and four exponentials conjecture
 Lehmer's conjecture
 Pompeiu problem
 Are (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?^{[13]}^{[14]}^{[15]}
 Khabibullin's conjecture on integral inequalities
 Hilbert's thirteenth problem
 Vitushkin's conjecture
Combinatorics
 Frankl's unionclosed 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 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?^{[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 nstep selfavoiding 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 geometry
Discrete geometry
 Solving the happy ending problem for arbitrary ^{[23]}
 Finding matching upper and lower bounds for ksets and halving lines^{[24]}
 The Hadwiger conjecture on covering ndimensional convex bodies with at most 2^{n} 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 worstpacking 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 geometry
 The einstein problem – does there exist a twodimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?^{[31]}
 Inscribed square problem – does every Jordan curve have an inscribed square?^{[32]}
 Kakeya conjecture
 Moser's worm problem – what is the smallest area of a shape that can cover every unitlength curve in the plane?^{[33]}
 The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unitwidth Lshaped corridor?^{[34]}
 Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net?^{[35]}
 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)?
 Falconer's conjecture
 gconjecture
 Circle packing in an equilateral triangle
 Circle packing in an isosceles right triangle
 Is 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 5polytopes^{[36]}
 Covering problem of Rado
 The strong bellows conjecture – must the Dehn invariant of a selfintersection free flexible polyhedron stay constant as it flexes?
Dynamical systems
 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 ergodic measure for the action on the circle either Lebesgue or atomic?
 Margulis conjecture – Measure classification for diagonalizable actions in higherrank groups
 MLC 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?
 Is every reversible cellular automaton in three or more dimensions locally reversible?^{[37]}
 Many problems concerning an outer billiard, for example show that outer billiards relative to almost every convex polygon has unbounded orbits.
Games and puzzles
 Chess (finding an optimal strategy for playing chess)
 Sudoku (what is the maximum number of givens for a minimal puzzle?)^{[38]}
 Sudoku (can every Sudoku be solved by logic, other than a bruteforce analysis?)^{[38]}
 Sudoku (how many proper puzzles exist, i.e. puzzles with one solution?)^{[38]}
 Sudoku (how many minimal proper puzzles exist, i.e. minimal puzzles with one solution?)^{[38]}
Graph theory
Paths and cycles in graphs
 Barnette's conjecture that every cubic bipartite threeconnected planar graph has a Hamiltonian cycle^{[39]}
 Chvátal's toughness conjecture, that there is a number t such that every ttough graph is Hamiltonian^{[40]}
 The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice^{[41]}
 The Erdős–Gyárfás conjecture on cycles with poweroftwo lengths in cubic graphs^{[42]}
 The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree^{[43]}
 The Lovász conjecture on Hamiltonian paths in symmetric graphs^{[44]}
Graph coloring and labeling
 The Erdős–Faber–Lovász conjecture on coloring unions of cliques^{[45]}
 The Hadwiger conjecture relating coloring to clique minors^{[46]}
 The Hadwiger–Nelson problem on the chromatic number of unit distance graphs^{[47]}
 Hedetniemi's conjecture on the chromatic number of tensor products of graphs^{[48]}
 Jaeger's Petersencoloring conjecture that every bridgeless cubic graph has a cyclecontinuous mapping to the Petersen graph^{[49]}
 The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index^{[50]}
 The Ringel–Kotzig conjecture on graceful labeling of trees^{[51]}
 The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree^{[52]}
Graph drawing
 The Albertson conjecture that the crossing number can be lowerbounded by the crossing number of a complete graph with the same chromatic number^{[53]}
 The Blankenship–Oporowski conjecture on the book thickness of subdivisions^{[54]}
 Conway's thrackle conjecture^{[55]}
 Harborth's conjecture that every planar graph can be drawn with integer edge lengths^{[56]}
 Negami's conjecture on projectiveplane embeddings of graphs with planar covers^{[57]}
 The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding^{[58]}
 Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?^{[59]}
 Universal point sets of subquadratic size for planar graphs^{[60]}
Miscellaneous graph theory
 The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph^{[61]}
 The implicit graph conjecture on the existence of implicit representations for slowlygrowing hereditary families of graphs^{[62]}
 Jørgensen's conjecture that every 6vertexconnected K_{6}minorfree graph is an apex graph^{[63]}
 Deriving a closedform expression for the percolation threshold values, especially (square site)
 Does a Moore graph with girth 5 and degree 57 exist?
 What is the largest possible pathwidth of an nvertex cubic graph?
 The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertexdeleted subgraphs.
 Sumner's conjecture: does every vertex tournament contain as a subgraph every vertex oriented tree?^{[64]}
 Tutte's conjectures that every bridgeless graph has a nowherezero 5flow and every Petersenminorfree bridgeless graph has a nowherezero 4flow
 Vizing's conjecture on the domination number of cartesian products of graphs^{[65]}
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?
 Andrews–Curtis conjecture
 Herzog–Schönheim conjecture
 Does generalized moonshine exist?
 Are there an infinite number of Leinster Groups?
Model theory
 Vaught's conjecture
 The Cherlin–Zilber conjecture: A simple group whose firstorder 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.^{[66]}
 Determine the structure of Keisler's order^{[67]}^{[68]}
 The stable field conjecture: every infinite field with a stable firstorder 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 wellordering consistently decidable?^{[69]}
 The Stable Forking Conjecture for simple theories^{[70]}
 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?^{[71]}
 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 .^{[66]}^{[72]}
 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.^{[66]}
 Is there a logic L which satisfies both the Beth property and Δinterpolation, is compact but does not satisfy the interpolation property?^{[73]}
 If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?^{[74]}^{[75]}
 Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
 Kueker's conjecture^{[76]}
 Does there exist an ominimal first order theory with a transexponential (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. trianglefree graphs)
 The universality problem for Cfree graphs: For which finite sets C of graphs does the class of Cfree countable graphs have a universal member under strong embeddings?^{[77]}
 The universality spectrum problem: Is there a firstorder theory whose universality spectrum is minimum?^{[78]}
Number theory
General
 Grand Riemann hypothesis

n conjecture
 abc conjecture (Proof claimed in 2012, currently under review.)
 Hilbert's ninth problem
 Hilbert's eleventh problem
 Hilbert's twelfth problem
 Carmichael's totient function conjecture
 Erdős–Straus conjecture
 Pillai's conjecture
 Hall's conjecture
 Lindelöf hypothesis
 Montgomery's pair correlation conjecture
 Hilbert–Pólya conjecture
 Grimm's conjecture
 Leopoldt's 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?
 Catalan–Dickson conjecture on aliquot sequences
 Do any Taxicab(5, 2, n) exist for n > 1?
 Brocard's problem: existence of integers, (n,m), such that n! + 1 = m^{2} other than n = 4, 5, 7
 Beilinson conjecture
 Littlewood conjecture
 Szpiro's conjecture
 Vojta's conjecture
 Goormaghtigh conjecture
 Congruent number problem (a corollary to Birch and SwinnertonDyer 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 problem
 Exponent pair conjecture
 Is π a normal number (its digits are "random")?^{[79]}
 CasasAlvero conjecture
 Sato–Tate conjecture
 Find value of De Bruijn–Newman constant
 Which integers can be written as the sum of three perfect cubes?^{[80]}
 Erdős–Moser problem: is 1^{1} + 2^{1} = 3^{1} the only solution to the Erdős–Moser equation?
 Is there a covering system with odd distinct moduli?^{[81]}
Additive number theory
 Beal's conjecture
 Fermat–Catalan conjecture
 Goldbach's conjecture
 The values of g(k) and G(k) in Waring's problem
 Lander, Parkin, and Selfridge conjecture
 Gilbreath's conjecture
 Erdős conjecture on arithmetic progressions
 Erdős–Turán conjecture on additive bases
 Pollock octahedral numbers conjecture
 Skolem problem
 Determine growth rate of r_{k}(N) (see Szemerédi's theorem)
 Minimum overlap problem
 Do the Ulam numbers have a positive density?
Algebraic number theory
 Are 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 conjecture
Combinatorial number theory
 Singmaster'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?
Prime numbers
 Catalan's Mersenne conjecture
 Agoh–Giuga 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?
 New Mersenne conjecture
 Erdős–Mollin–Walsh conjecture
 Are 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 −4k^{4} 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 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 squarefree?
 Are there infinitely many Wieferich primes?
 Are there any Wieferich primes in base 47?
 Are there any composite c satisfying 2^{c − 1} ≡ 1 (mod c^{2})?
 For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p^{2})?^{[82]}
 Can a prime p satisfy 2^{p − 1} ≡ 1 (mod p^{2}) and 3^{p − 1} ≡ 1 (mod p^{2}) simultaneously?^{[83]}
 Are there infinitely many Wilson primes?
 Are there infinitely many Wolstenholme primes?
 Are there any Wall–Sun–Sun primes?
 Is every Fermat number 2^{2n} + 1 composite for ?
 Are all Fermat numbers squarefree?
 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 roots
 Is 78,557 the lowest Sierpiński number (socalled Selfridge's conjecture)?
 Is 509,203 the lowest Riesel number?
 Fortune's conjecture (that no Fortunate number is composite)
 Landau's problems
 Feit–Thompson conjecture
 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
 Problems associated to Linnik's theorem
 Find the smallest Skewes' number
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
 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^{ℵω} < ℵ_{ω1} (see Singular cardinals hypothesis). The best bound, ℵ_{ω4}, was obtained by Shelah using his pcf theory.
 Woodin's Ωhypothesis.
 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 ℵ_{ω}?
 Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
 Does the Generalized Continuum Hypothesis entail for every singular cardinal ?
 Does the Generalized Continuum Hypothesis imply the existence of an ℵ_{2}Suslin tree?
 Is OCA (Open coloring axiom) consistent with ?
Other
 List of unsolved problems in statistics
 List of unsolved problems in computer science
 List of unsolved problems in physics
 Problems in loop theory and quasigroup theory
 Problems in Latin squares
 Invariant subspace problem
 Kaplansky's conjectures on groups rings
 Painlevé conjecture
 Dixmier conjecture
 Baum–Connes conjecture
 Novikov conjecture
 Prove Turing completeness for all unique elementary cellular automaton
 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.^{[84]}
 Keating–Snaith conjecture
 Kung–Traub conjecture
 Atiyah conjecture on configurations
 Toeplitz' conjecture (open since 1911)
 Carathéodory conjecture
 Weightmonodromy conjecture
 Berry–Tabor conjecture
 Birkhoff conjecture
 Guralnick–Thompson conjecture
 Hilbert–Smith conjecture
 MNOP conjecture
 Mazur's conjectures
 Rendezvous problem
 Scholz conjecture
 Nirenberg–Treves conjecture
 Quantum unique ergodicity conjecture
 Density hypothesis
 Zhou conjecture
 Borel conjecture
 Zeeman conjecture
 Erdős–Ulam problem
Problems solved since 1995
 Pentagonal tiling (Michaël Rao, 2017)^{[85]}
 Erdős–Burr conjecture (Choongbum Lee, 2017)^{[86]}
 Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)^{[87]}
 Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (A. Abdollahi, M. Zallaghi, 2015)^{[88]}
 Main conjecture in Vinogradov's meanvalue theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)^{[89]}
 Erdős discrepancy problem (Terence Tao, 2015)^{[90]}
 Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)^{[91]}
 Anderson conjecture (Cheeger, Naber, 2014)^{[92]}
 Gaussian correlation inequality (Thomas Royen, 2014)^{[93]}
 Goldbach's weak conjecture (Harald Helfgott, 2013)^{[94]}^{[95]}^{[96]}
 Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)^{[97]}^{[98]} (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic and conjectures, BourgainTzafriri conjecture and conjecture)
 Virtual Haken conjecture (Agol, Groves, Manning, 2012)^{[99]} (and by work of Wise also virtually fibered conjecture)
 Hsiang–Lawson's conjecture (Brendle, 2012)^{[100]}
 Willmore conjecture (Fernando Codá Marques and André Neves, 2012)^{[101]}
 Ehrenpreis conjecture (Kahn, Markovic, 2011)^{[102]}
 Hanna Neumann conjecture (Mineyev, 2011)^{[103]}
 Bloch–Kato conjecture (Voevodsky, 2011)^{[104]} (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture^{[105]}^{[106]}^{[107]})
 Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)^{[108]}
 Density theorem (Namazi, Souto, 2010)^{[109]}
 Hirsch conjecture (Francisco Santos Leal, 2010)^{[110]}^{[111]}
 Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)^{[112]}
 Atiyah conjecture (Austin, 2009)^{[113]}
 Kauffman–Harary conjecture (Matmann, Solis, 2009)^{[114]}
 Surface subgroup conjecture (Kahn, Markovic, 2009)^{[115]}
 Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)^{[116]}
 Cobordism hypothesis (Jacob Lurie, 2008)^{[117]}
 Full classification of finite simple groups (Harada, Solomon, 2008)
 Geometrization conjecture (proof was completed by Morgan and Tian in 2008^{[118]} and it is based mostly on work of Grigori Perelman, 2002)^{[119]}
 Serre's modularity conjecture (Chandrashekhar Khare and JeanPierre Wintenberger, 2008)^{[120]}^{[121]}^{[122]}
 Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)^{[123]}
 Erdős–Menger conjecture (Aharoni, Berger 2007)^{[124]}
 Road coloring conjecture (Avraham Trahtman, 2007)^{[125]}
 The angel problem (Various independent proofs, 2006)^{[126]}^{[127]}^{[128]}^{[129]}
 Lax conjecture (Lewis, Parrilo, Ramana, 2005)^{[130]}
 The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)^{[131]}
 Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)^{[132]}
 Robertson–Seymour theorem (Robertson, Seymour, 2004)^{[133]}
 Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)^{[134]} (and also Alon–Friedgut conjecture)
 Green–Tao theorem (Ben J. Green and Terence Tao, 2004)^{[135]}
 Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)^{[136]}
 Carpenter's rule problem (Connelly, Demaine, Rote, 2003)^{[137]}
 Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)^{[138]}^{[139]}
 Milnor conjecture (Vladimir Voevodsky, 2003)^{[140]}
 Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)^{[141]}
 Nagata's conjecture (Shestakov, Umirbaev, 2003)^{[142]}
 Kirillov's conjecture (Baruch, 2003)^{[143]}
 Poincaré conjecture (Grigori Perelman, 2002)^{[119]}
 Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)^{[144]}
 Kouchnirenko’s conjecture (Haas, 2002)^{[145]}
 Vaught conjecture (Knight, 2002)^{[146]}
 Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)^{[147]}
 Catalan's conjecture (Preda Mihăilescu, 2002)^{[148]}
 n! conjecture (Haiman, 2001)^{[149]} (and also Macdonald positivity conjecture)
 Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)^{[150]}
 Deligne's conjecture on 1motives (Luca BarbieriViale, Andreas Rosenschon, Morihiko Saito, 2001)^{[151]}
 Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)^{[152]}
 Erdős–Stewart conjecture (Florian Luca, 2001)^{[153]}
 Berry–Robbins problem (Atiyah, 2000)^{[154]}
 Erdős–Graham problem (Croot, 2000)^{[155]}
 Honeycomb conjecture (Thomas Hales, 1999)^{[156]}
 Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)^{[157]}
 Bogomolov conjecture (Emmanuel Ullmo, 1998, ShouWu Zhang, 1998)^{[158]}^{[159]}
 Lafforgue's theorem (Laurent Lafforgue, 1998)^{[160]}
 Kepler conjecture (Ferguson, Hales, 1998)^{[161]}
 Dodecahedral conjecture (Hales, McLaughlin, 1998)^{[162]}
 Ganea conjecture (Iwase, 1997)^{[163]}
 Torsion conjecture (Merel, 1996)^{[164]}
 Harary's conjecture (Chen, 1996)^{[165]}
 Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)^{[166]}^{[167]}
References
 ^ Eves, An Introduction to the History of Mathematics 6th Edition, Thomson, 1990, ISBN 9780030295584.
 ^ Thiele, Rüdiger (2005), "On Hilbert and his twentyfour problems", in Van Brummelen, Glen, Mathematics and the historian's craft. The Kenneth O. May Lectures, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 21, pp. 243–295, ISBN 0387252843
 ^ Guy, Richard (1994), Unsolved Problems in Number Theory (2nd ed.), Springer, p. vii, ISBN 9781489935854.
 ^ Shimura, G. (1989). "Yutaka Taniyama and his time". Bulletin of the London Mathematical Society. 21 (2): 186–196. doi:10.1112/blms/21.2.186.
 ^ http://www.uniregensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/dmv_091514.pdf
 ^ "THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY" (PDF).
 ^ Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 9051994907.
 ^ "DARPA invests in math". CNN. 20081014. Archived from the original on 20090304. Retrieved 20130114.
 ^ "Broad Agency Announcement (BAA 0768) for Defense Sciences Office (DSO)". DARPA. 20070910. Archived from the original on 20121001. Retrieved 20130625.
 ^ "Millennium Problems".
 ^ "Poincaré Conjecture". Clay Mathematics Institute. Archived from the original on 20131215.
 ^ "Smooth 4dimensional Poincare conjecture".
 ^ For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi ([1]), e ([2]), Khinchin's Constant ([3]), irrational numbers ([4]), transcendental numbers ([5]), and irrationality measures ([6]) at Wolfram MathWorld, all articles accessed 15 December 2014.
 ^ Michel Waldschmidt, 2008, "An introduction to irrationality and transcendence methods," at The University of Arizona The Southwest Center for Arithmetic Geometry 2008 Arizona Winter School, March 15–19, 2008 (Special Functions and Transcendence), see [7], accessed 15 December 2014.
 ^ John Albert, posting date unknown, "Some unsolved problems in number theory" [from Victor Klee & Stan Wagon, "Old and New Unsolved Problems in Plane Geometry and Number Theory"], in University of Oklahoma Math 4513 course materials, see [8], accessed 15 December 2014.
 ^ Bruhn, Henning; Schaudt, Oliver (2015), "The journey of the unionclosed sets conjecture" (PDF), Graphs and Combinatorics, 31 (6): 2043–2074, doi:10.1007/s0037301415150, MR 3417215
 ^ Tao, Terence (2017), Some remarks on the lonely runner conjecture, arXiv:1701.02048
 ^ Singmaster, D. (1971), "Research Problems: How often does an integer occur as a binomial coefficient?", American Mathematical Monthly, 78 (4): 385–386, doi:10.2307/2316907, JSTOR 2316907, MR 1536288.
 ^ Liśkiewicz, Maciej; Ogihara, Mitsunori; Toda, Seinosuke (20030728). "The complexity of counting selfavoiding walks in subgraphs of twodimensional grids and hypercubes". Theoretical Computer Science. 304 (1): 129–156. doi:10.1016/S03043975(03)00080X.
 ^ Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, doi:10.1007/BF01110378, MR 1368815.
 ^ Aigner, Martin (2013), Markov's theorem and 100 years of the uniqueness conjecture, Cham: Springer, doi:10.1007/9783319008882, ISBN 9783319008875, MR 3098784
 ^ Murnaghan, F. D. (1938), "The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups", American Journal of Mathematics, 60 (1): 44–65, doi:10.2307/2371542, MR 1507301
 ^ Morris, Walter D.; Soltan, Valeriu (2000), "The ErdősSzekeres problem on points in convex position—a survey", Bull. Amer. Math. Soc., 37 (4): 437–458, doi:10.1090/S0273097900008776, MR 1779413; Suk, Andrew (2016), "On the Erdős–Szekeres convex polygon problem", J. Amer. Math. Soc., 30: 1047–1053, arXiv:1604.08657 , doi:10.1090/jams/869
 ^ Dey, Tamal K. (1998), "Improved bounds for planar ksets and related problems", Discrete Comput. Geom., 19: 373–382, doi:10.1007/PL00009354, MR 1608878; Tóth, Gábor (2001), "Point sets with many ksets", Discrete Comput. Geom., 26 (2): 187–194, doi:10.1007/s004540010022, MR 1843435.
 ^ Boltjansky, V.; Gohberg, I. (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
 ^ Weisstein, Eric W. "Kobon Triangle". MathWorld.
 ^ Matoušek, Jiří (2002), Lectures on discrete geometry, Graduate Texts in Mathematics, 212, SpringerVerlag, New York, p. 206, doi:10.1007/9781461300397, ISBN 0387953736, MR 1899299
 ^ Gardner, Martin (1995), New Mathematical Diversions (Revised Edition), Washington: Mathematical Association of America, p. 251
 ^ Conway, John H.; Neil J.A. Sloane (1999), Sphere Packings, Lattices and Groups (3rd ed.), New York: SpringerVerlag, pp. 21–22, ISBN 0387985859
 ^ Brass, Peter; Moser, William; Pach, János (2005), "5.1 The Maximum Number of Unit Distances in the Plane", Research problems in discrete geometry, Springer, New York, pp. 183–190, ISBN 0387238158, MR 2163782
 ^ Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419 , doi:10.1007/s002830119255y, MR 2902144
 ^ Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100
 ^ Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete and Computational Geometry, 7 (2): 153–162, doi:10.1007/BF02187832, MR 1139077
 ^ Wagner, Neal R. (1976), "The Sofa Problem" (PDF), The American Mathematical Monthly, 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022
 ^ Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338
 ^ ACW (May 24, 2012), "Convex uniform 5polytopes", Open Problem Garden, retrieved 20161004.
 ^ Kari, Jarkko (2009), "Structure of reversible cellular automata", Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7ÔÇô11, 2009, Proceedings, Lecture Notes in Computer Science, 5715, Springer, p. 6, doi:10.1007/9783642037450_5
 ^ ^{a} ^{b} ^{c} ^{d} http://english.logitex.com Ten open questions about Sudoku (20120121).
 ^ Florek, Jan (2010), "On Barnette's conjecture", Discrete Mathematics, 310 (10–11): 1531–1535, doi:10.1016/j.disc.2010.01.018, MR 2601261.
 ^ Broersma, Hajo; Patel, Viresh; Pyatkin, Artem (2014), "On toughness and Hamiltonicity of $2K_2$free graphs", Journal of Graph Theory, 75 (3): 244–255, doi:10.1002/jgt.21734, MR 3153119
 ^ Jaeger, F. (1985), "A survey of the cycle double cover conjecture", Annals of Discrete Mathematics 27 – Cycles in Graphs, NorthHolland Mathematics Studies, 27, pp. 1–12, doi:10.1016/S03040208(08)729931.
 ^ Heckman, Christopher Carl; Krakovski, Roi (2013), "ErdösGyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7.
 ^ Akiyama, Jin; Exoo, Geoffrey; Harary, Frank (1981), "Covering and packing in graphs. IV. Linear arboricity", Networks, 11 (1): 69–72, doi:10.1002/net.3230110108, MR 0608921.
 ^ L. Babai, Automorphism groups, isomorphism, reconstruction, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447–1540.
 ^ Chung, Fan; Graham, Ron (1998), Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, pp. 97–99.
 ^ Toft, Bjarne (1996), "A survey of Hadwiger's conjecture", Congressus Numerantium, 115: 249–283, MR 1411244.
 ^ Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991), Unsolved Problems in Geometry, SpringerVerlag, Problem G10.
 ^ Sauer, N. (2001), "Hedetniemi's conjecture: a survey", Discrete Mathematics, 229 (1–3): 261–292, doi:10.1016/S0012365X(00)002132, MR 1815610.
 ^ Hägglund, Jonas; Steffen, Eckhard (2014), "Petersencolorings and some families of snarks", Ars Mathematica Contemporanea, 7 (1): 161–173, MR 3047618.
 ^ Jensen, Tommy R.; Toft, Bjarne (1995), "12.20 ListEdgeChromatic Numbers", Graph Coloring Problems, New York: WileyInterscience, pp. 201–202, ISBN 0471028657.
 ^ Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 0668845.
 ^ Molloy, Michael; Reed, Bruce (1998), "A bound on the total chromatic number", Combinatorica, 18 (2): 241–280, doi:10.1007/PL00009820, MR 1656544.
 ^ Barát, János; Tóth, Géza (2010), "Towards the Albertson Conjecture", Electronic Journal of Combinatorics, 17 (1): R73, arXiv:0909.0413 .
 ^ Wood, David (January 19, 2009), "Book Thickness of Subdivisions", Open Problem Garden, retrieved 20130205.
 ^ Fulek, R.; Pach, J. (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry, 44 (6–7): 345–355, doi:10.1007/9783642184697_21, MR 2785903.
 ^ Hartsfield, Nora; Ringel, Gerhard (2013), Pearls in Graph Theory: A Comprehensive Introduction, Dover Books on Mathematics, Courier Dover Publications, p. 247, ISBN 9780486315522, MR 2047103.
 ^ Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, doi:10.1007/s0037301009349, MR 2669457.
 ^ Nöllenburg, Martin; Prutkin, Roman; Rutter, Ignaz (2016), "On selfapproaching and increasingchord drawings of 3connected planar graphs", Journal of Computational Geometry, 7 (1): 47–69, doi:10.20382/jocg.v7i1a3, MR 3463906
 ^ Pach, János; Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, 152, American Mathematical Society, pp. 126–127.
 ^ Demaine, E.; O'Rourke, J. (2002–2012), "Problem 45: Smallest Universal Set of Points for Planar Graphs", The Open Problems Project, retrieved 20130319.
 ^ Chudnovsky, Maria (2014), "The Erdös–Hajnal conjecture—a survey" (PDF), Journal of Graph Theory, 75 (2): 178–190, arXiv:1606.08827 , doi:10.1002/jgt.21730, MR 3150572, Zbl 1280.05086.
 ^ Spinrad, Jeremy P. (2003), "2. Implicit graph representation", Efficient Graph Representations, pp. 17–30, ISBN 0821828150.
 ^ "Jorgensen's Conjecture", Open Problem Garden, retrieved 20161113.
 ^ Kühn, Daniela; Mycroft, Richard; Osthus, Deryk (2011), "A proof of Sumner's universal tournament conjecture for large tournaments", Proceedings of the London Mathematical Society, Third Series, 102 (4): 731–766, arXiv:1010.4430 , doi:10.1112/plms/pdq035, MR 2793448, Zbl 1218.05034.
 ^ Brešar, Boštjan; Dorbec, Paul; Goddard, Wayne; Hartnell, Bert L.; Henning, Michael A.; Klavžar, Sandi; Rall, Douglas F. (2012), "Vizing's conjecture: a survey and recent results", Journal of Graph Theory, 69 (1): 46–76, doi:10.1002/jgt.20565, MR 2864622.
 ^ ^{a} ^{b} ^{c} Shelah S, Classification Theory, NorthHolland, 1990
 ^ Keisler, HJ, "Ultraproducts which are not saturated." J. Symb Logic 32 (1967) 23—46.
 ^ Malliaris M, Shelah S, "A dividing line in simple unstable theories." http://arxiv.org/abs/1208.2140
 ^ Gurevich, Yuri, "Monadic SecondOrder Theories," in J. Barwise, S. Feferman, eds., ModelTheoretic Logics (New York: SpringerVerlag, 1985), 479–506.
 ^ Peretz, Assaf, "Geometry of forking in simple theories." J. Symbolic Logic Volume 71, Issue 1 (2006), 347–359.
 ^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae. 159 (1): 1–50. arXiv:math/9802134 .
 ^ Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN 9781904987710.
 ^ Makowsky J, "Compactness, embeddings and definability," in ModelTheoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
 ^ Baldwin, John T. (July 24, 2009). Categoricity (PDF). American Mathematical Society. ISBN 9780821848937. Retrieved February 20, 2014.
 ^ Shelah, Saharon. "Introduction to classification theory for abstract elementary classes".
 ^ Hrushovski, Ehud (1989). "Kueker's conjecture for stable theories". Journal of Symbolic Logic. 54 (1): 207–220. doi:10.2307/2275025.
 ^ Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B. 97 (3): 293–333. arXiv:math/0512218 . doi:10.1016/j.jctb.2006.05.008.
 ^ Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
 ^ "Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key".
 ^ http://arxiv.org/pdf/1604.07746v1.pdf
 ^ Guo, Song; Sun, ZhiWei (2005), "On odd covering systems with distinct moduli", Advances in Applied Mathematics, 35 (2): 182–187, doi:10.1016/j.aam.2005.01.004, MR 2152886
 ^ Ribenboim, P. (2006). Die Welt der Primzahlen (in German) (2nd ed.). Springer. pp. 242–243. doi:10.1007/9783642180798. ISBN 9783642180781.
 ^ Dobson, J. B. (June 2012) [2011], On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907
 ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society, 125: 1503–1509, JSTOR 2162098
 ^ Wolchover, Natalie (July 11, 2017), "Pentagon Tiling Proof Solves CenturyOld Math Problem", Quanta Magazine
 ^ Lee, Choongbum (2017), "Ramsey numbers of degenerate graphs", Annals of Mathematics, 185 (3): 791–829, doi:10.4007/annals.2017.185.3.2
 ^ Lamb, Evelyn (26 May 2016), "Twohundredterabyte maths proof is largest ever", Nature, 534: 17–18, doi:10.1038/nature.2016.19990, PMID 27251254; Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016), "Solving and verifying the Boolean Pythagorean triples problem via cubeandconquer", Theory and applications of satisfiability testing—SAT 2016, Lecture Notes in Comput. Sci., 9710, Springer, [Cham], pp. 228–245, arXiv:1605.00723 , doi:10.1007/9783319409702_15, MR 3534782
 ^ Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398.
 ^ Bourgain, Jean; Ciprian, Demeter; Larry, Guth (2015). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". Annals of Mathematics. doi:10.4007/annals.2016.184.2.7.
 ^ http://arxiv.org/pdf/1509.05363v5.pdf
 ^ Duncan, John F. R.; Griffin, Michael J.; Ono, Ken (1 December 2015). "Proof of the umbral moonshine conjecture". Research in the Mathematical Sciences. 2 (1): 26. doi:10.1186/s4068701500447 – via link.springer.com.
 ^ http://arxiv.org/pdf/1406.6534v10.pdf
 ^ "A LongSought Proof, Found and Almost Lost". Quanta Magazine. Natalie Wolchover. March 28, 2017. Retrieved May 2, 2017.
 ^ Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
 ^ Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
 ^ Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
 ^ Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric (2006). Han, Deguang; Jorgensen, Palle E. T.; Larson, David Royal, eds. "The KadisonSinger problem in mathematics and engineering: A detailed account". Contemporary Mathematics. Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida. American Mathematical Society. 414: 299–355. doi:10.1090/conm/414/07820. ISBN 9780821839232. Retrieved 24 April 2015.
 ^ Mackenzie, Dana. "Kadison–Singer Problem Solved" (PDF). SIAM News (January/February 2014). Society for Industrial and Applied Mathematics. Retrieved 24 April 2015.
 ^ http://arxiv.org/pdf/1204.2810v1.pdf
 ^ http://www.math.jhu.edu/~js/Math646/brendle.lawson.pdf
 ^ Marques, Fernando C.; Neves, André (2013). "Minmax theory and the Willmore conjecture". Annals of Mathematics. 179: 683–782. arXiv:1202.6036 . doi:10.4007/annals.2014.179.2.6.
 ^ http://arxiv.org/pdf/1101.1330v4.pdf
 ^ http://www.math.uiuc.edu/~mineyev/math/art/submultshnc.pdf
 ^ http://annals.math.princeton.edu/wpcontent/uploads/annalsv174n1p11p.pdf
 ^ https://www.unidue.de/~bm0032/publ/BlochKato.pdf
 ^ "page 359" (PDF).
 ^ "motivic cohomology – Milnor–Bloch–Kato conjecture implies the BeilinsonLichtenbaum conjecture – MathOverflow".
 ^ http://arxiv.org/pdf/1011.4105v3.pdf
 ^ https://www.researchgate.net/profile/Juan_Souto3/publication/228365532_Nonrealizability_and_ending_laminations_Proof_of_the_Density_Conjecture/links/541d85a10cf2218008d1d2e5.pdf
 ^ Santos, Franciscos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics. Princeton University and Institute for Advanced Study. 176 (1): 383–412. doi:10.4007/annals.2012.176.1.7.
 ^ Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica. Extra Volume "Optimization Stories": 75–85.
 ^ "Generalized Sidon sets". Advances in Mathematics. 225: 2786–2807. doi:10.1016/j.aim.2010.05.010.
 ^ http://arxiv.org/pdf/0909.2360v3.pdf
 ^ http://arxiv.org/pdf/0906.1612v2.pdf
 ^ http://arxiv.org/pdf/0910.5501v5.pdf
 ^ http://www.csie.ntu.edu.tw/~hil/bib/ChalopinG09.pdf
 ^ Lurie, Jacob. "On the classification of topological field theories." Current developments in mathematics 2008 (2009): 129280.
 ^ http://arxiv.org/pdf/0809.4040.pdf

^ ^{a} ^{b} "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF) (Press release). Clay Mathematics Institute. March 18, 2010. Retrieved November 13, 2015.
The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.
 ^ Khare, Chandrashekhar; Wintenberger, JeanPierre (2009), "Serre's modularity conjecture (I)", Inventiones Mathematicae, 178 (3): 485–504, doi:10.1007/s0022200902057
 ^ Khare, Chandrashekhar; Wintenberger, JeanPierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505–586, doi:10.1007/s0022200902066
 ^ "2011 Cole Prize in Number Theory" (PDF). Notices of the AMS. Providence, Rhode Island, United States: American Mathematical Society. 58 (4): 610–611. ISSN 10889477. OCLC 34550461.
 ^ http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf
 ^ http://arxiv.org/pdf/math/0509397.pdf
 ^ SeigelItzkovich, Judy (20080208). "Russian immigrant solves math puzzle". The Jerusalem Post. Retrieved 20151112.
 ^ http://homepages.warwick.ac.uk/~masgak/papers/bhbangel.pdf
 ^ http://home.broadpark.no/~oddvark/angel/Angel.pdf
 ^ http://homepages.warwick.ac.uk/~masibe/angelmathe.pdf
 ^ http://www.cs.bu.edu/~gacs/papers/angel.pdf
 ^ http://www.ams.org/journals/proc/200513309/S000299390507752X/S000299390507752X.pdf

^ "Fields Medal – Ngô Bảo Châu". International Congress of Mathematicians 2010. ICM. 19 August 2010. Retrieved 20151112.
Ngô Bảo Châu is being awarded the 2010 Fields Medal for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebrogeometric methods.
 ^ http://arxiv.org/pdf/math/0405568v1.pdf
 ^ "Graph Theory".

^ Chung, Fan; Greene, Curtis; Hutchinson, Joan (April 2015). "Herbert S. Wilf (1931–2012)" (PDF). Notices of the AMS. Providence, Rhode Island, United States: American Mathematical Society. 62 (4): 358. ISSN 10889477. OCLC 34550461.
The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004.

^ "Bombieri and Tao Receive King Faisal Prize" (PDF). Notices of the AMS. Providence, Rhode Island, United States: American Mathematical Society. 57 (5): 642–643. May 2010. ISSN 10889477. OCLC 34550461.
Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem.
 ^ http://arxiv.org/pdf/math/0412006v2.pdf
 ^ Connelly, Robert; Demaine, Erik D.; Rote, Günter (2003), "Straightening polygonal arcs and convexifying polygonal cycles" (PDF), Discrete and Computational Geometry, 30 (2): 205–239, doi:10.1007/s0045400300067, MR 1931840
 ^ Green, Ben (2004), "The Cameron–Erdős conjecture", The Bulletin of the London Mathematical Society, 36 (6): 769–778, arXiv:math.NT/0304058 , doi:10.1112/S0024609304003650, MR 2083752

^ "News from 2007". American Mathematical Society. AMS. 31 December 2007. Retrieved 20151113.
The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the CameronErdős conjecture..."
 ^ "Reduced power operations in motivic cohomology" (PDF). archive.numdam.org.
 ^ "Kemnitz' conjecture revisited". Discrete Mathematics. 297: 196–201. doi:10.1016/j.disc.2005.02.018.
 ^ http://www.ams.org/journals/jams/20041701/S0894034703004405/S0894034703004405.pdf
 ^ http://annals.math.princeton.edu/wpcontent/uploads/annalsv158n1p04.pdf
 ^ "The strong perfect graph theorem".
 ^ http://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf
 ^ Knight, R. W. (2002), The Vaught Conjecture: A Counterexample, manuscript
 ^ http://www.ugr.es/~ritore/preprints/0406017.pdf

^ Metsänkylä, Tauno (5 September 2003). "Catalan's conjecture: another old diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. American Mathematical Society. 41 (1): 43–57. doi:10.1090/s0273097903009935. ISSN 02730979.
The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu.
 ^ http://www.ams.org/journals/jams/20011404/S0894034701003733/S0894034701003733.pdf
 ^ http://junon.u3mrs.fr/monniaux/AHLMT02.pdf
 ^ http://arxiv.org/pdf/math/0102150v4.pdf
 ^ Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3adic exercises", Journal of the American Mathematical Society, 14 (4): 843–939, doi:10.1090/S0894034701003708, ISSN 08940347, MR 1839918
 ^ http://www.ams.org/journals/mcom/200170234/S0025571800011789/S0025571800011789.pdf
 ^ http://intlpress.com/site/pub/files/_fulltext/journals/sdg/2002/0007/0001/SDG200200070001a001.pdf
 ^ Croot, Ernest S., III (2000), Unit Fractions, Ph.D. thesis, University of Georgia, Athens. Croot, Ernest S., III (2003), "On a coloring conjecture about unit fractions", Annals of Mathematics, 157 (2): 545–556, arXiv:math.NT/0311421 , doi:10.4007/annals.2003.157.545
 ^ http://arxiv.org/pdf/math/9906042v2.pdf
 ^ http://arxiv.org/pdf/math/9906212v2.pdf
 ^ Ullmo, E (1998). "Positivité et Discrétion des Points Algébriques des Courbes". Annals of Mathematics. 147 (1): 167–179. doi:10.2307/120987. Zbl 0934.14013.
 ^ Zhang, S.W. (1998). "Equidistribution of small points on abelian varieties". Annals of Mathematics. 147 (1): 159–165. doi:10.2307/120986.
 ^ Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 14310635, MR 1648105
 ^ http://arxiv.org/pdf/1501.02155.pdf
 ^ http://arxiv.org/pdf/math/9811079v3.pdf
 ^ Norio Iwase (1 November 1998). "Ganea's Conjecture on LusternikSchnirelmann Category". ResearchGate.
 ^ Merel, Loïc (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]. Inventiones Mathematicae (in French) 124 (1): 437–449. doi:10.1007/s002220050059. MR1369424
 ^ https://www.researchgate.net/profile/Zhibo_Chen/publication/220188021_Harary's_conjectures_on_integral_sum_graphs/links/5422b2490cf290c9e3aac7fe.pdf
 ^ Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. Annals of Mathematics. 141 (3): 443–551. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.
 ^ Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. Annals of Mathematics. 141 (3): 553–572. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255.
Further reading
Books discussing recently solved problems^{[needs update]}
 Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1841157910.
 O'Shea, Donal (2007). The Poincaré Conjecture. Penguin. ISBN 9781846140129.
 Szpiro, George G. (2003). Kepler's Conjecture. Wiley. ISBN 0471086010.
 Ronan, Mark (2006). Symmetry and the Monster. Oxford. ISBN 0192807226.
Books discussing unsolved problems
 Fan Chung; Graham, Ron (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 156881111X.
 Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 0387975063.
 Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 0387208607.
 Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0883853159.
 Du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0060935588.
 Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0309085497.
 Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 9780760786598.
 Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0691117489.
 Lizhen Ji, [various]; YatSun Poon, ShingTung Yau (2013). Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics). International Press of Boston. ISBN 1571462783.
 Waldschmidt, Michel (2004). "Open Diophantine Problems" (PDF). Moscow Mathematical Journal. 4 (1): 245–305. ISSN 16093321. Zbl 1066.11030.
 Mazurov, V. D.; Khukhro, E. I. (1 Jun 2015). "Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)". arXiv:1401.0300v6 .
 Derbyshire, John (2003). Prime Obsession. The Joseph Henry Press. ISBN 0309085497.
External links
 24 Unsolved Problems and Rewards for them
 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.
 Ball, John M. "Some Open Problems in Elasticity" (PDF).
 Constantin, Peter. "Some open problems and research directions in the mathematical study of fluid dynamics" (PDF).
 Serre, Denis. "Five Open Problems in Compressible Mathematical Fluid Dynamics" (PDF).
 Unsolved Problems in Number Theory, Logic and Cryptography
 200 open problems in graph theory
 The Open Problems Project (TOPP), discrete and computational geometry problems
 Kirby's list of unsolved problems in lowdimensional topology
 Erdös' Problems on Graphs
 A List of Approachable Open Problems in Knot Theory
 Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
 Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications
 List of open problems in inner model theory
 Aizenman, Michael. "Open Problems in Mathematical Physics".
 15 Problems in Mathematical Physics