List of unsolved problems in mathematics

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 long-standing 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.

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, six have yet to be solved, as of 2016:^[10]

• P versus NP
• Hodge conjecture
• Riemann hypothesis
• Yang–Mills existence and mass gap
• Navier–Stokes existence and smoothness
• Birch and Swinnerton-Dyer conjecture

The seventh problem, the Poincaré conjecture, has been solved.^[11] 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.^[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 SIC-POVMs 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 p-curvature conjecture
• Sendov's conjecture

Algebraic geometry

• André–Oort conjecture
• Bass conjecture
• Deligne conjecture
• Fröberg conjecture
• Fujita conjecture
• Hartshorne conjectures
• Manin conjecture
• Nakai conjecture
• Resolution of singularities in characteristic p
• Standard conjectures on algebraic cycles
• Section conjecture
• Tate conjecture
• Virasoro conjecture
• Whitehead conjecture
• Zariski multiplicity conjecture

Analysis

• The Jacobian conjecture
• Schanuel's conjecture and four exponentials conjecture
• Lehmer's conjecture
• Pompeiu problem
• Are $\gamma$ (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

• Number of magic squares (sequence A006052 in the OEIS)
• Number of magic tori (sequence A270876 in the OEIS)
• Finding a formula for the probability that two elements chosen at random generate the symmetric group $S_{n}$
• Frankl's union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
• The lonely runner conjecture: if $k+1$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance $1/(k+1)$ from each other runner) at some time?
• Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
• The 1/3–2/3 conjecture: does every finite partially ordered set 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?
• unicity conjecture for Markov numbers
• Give a combinatorial interpretation of the Kronecker coefficients.

Discrete geometry

• Solving the happy ending problem for arbitrary $n$
• Finding matching upper and lower bounds for k-sets and halving lines
• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2ⁿ smaller copies
• The Kobon triangle problem on triangles in line arrangements
• The McMullen problem on projectively transforming sets of points into convex position
• Ulam's packing conjecture about the identity of the worst-packing convex solid
• Filling area conjecture
• Hopf conjecture
• Kakeya conjecture
• Kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24
• How many unit distances can be determined by a set of n points? (see Counting unit distances)

Euclidean geometry

• The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?^[16]
• Inscribed square problem – does every Jordan curve have an inscribed square?^[17]
• Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?^[18]
• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?^[19]
• Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net?^[20]
• 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)?
• Pentagonal tiling – 15 types of convex pentagons are known to monohedrally tile the plane, and it is not known whether this list is complete.^[21]
• Falconer's conjecture
• g-conjecture
• Circle packing in an equilateral triangle
• Circle packing in an isosceles right triangle
• 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^[22]

Dynamical systems

• 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 $\times 2,\times 3$ action on the circle either Lebesgue or atomic?
• Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups
• MLC conjecture – Is the Mandelbrot set locally connected?
• 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?^[23]
• Many problems concerning an outer billiard, for example show that outer billiards relative to almost every convex polygon has unbounded orbits.

Graph theory

Paths and cycles in graphs

• Barnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle^[24]
• Chvátal's toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian^[25]
• The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice^[26]
• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs^[27]
• The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree^[28]
• The Lovász conjecture on Hamiltonian paths in symmetric graphs^[29]

Graph coloring and labeling

• The Erdős–Faber–Lovász conjecture on coloring unions of cliques^[30]
• The Hadwiger conjecture relating coloring to clique minors^[31]
• The Hadwiger–Nelson problem on the chromatic number of unit distance graphs^[32]
• Hedetniemi's conjecture on the chromatic number of tensor products of graphs^[33]
• Jaeger's Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph^[34]
• The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index^[35]
• The Ringel–Kotzig conjecture on graceful labeling of trees^[36]
• The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree^[37]

Graph drawing

• The Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number^[38]
• The Blankenship–Oporowski conjecture on the book thickness of subdivisions^[39]
• Conway's thrackle conjecture^[40]
• Harborth's conjecture that every planar graph can be drawn with integer edge lengths^[41]
• Negami's conjecture on projective-plane embeddings of graphs with planar covers^[42]
• The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding^[43]
• Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?^[44]
• Universal point sets of subquadratic size for planar graphs^[45]

Miscellaneous graph theory

• The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph^[46]
• Jørgensen's conjecture that every 6-vertex-connected K₆-minor-free graph is an apex graph^[47]
• Deriving a closed-form expression for the percolation threshold values, especially $p_{c}$ (square site)
• 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.
• Sumner's conjecture: does every $(2n-2)$-vertex tournament contain as a subgraph every $n$-vertex oriented tree?^[48]
• 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
• Vizing's conjecture on the domination number of cartesian products of graphs^[49]

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?

Model theory

• Vaught's conjecture
• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in $\aleph _{0}$ is a simple algebraic group over an algebraically closed field.
• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for $\aleph _{1}$-saturated models of a countable theory.^[50]
• Determine the structure of Keisler's order^[51][52]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• Is the theory of the field of Laurent series over $\mathbb {Z} _{p}$ decidable? of the field of polynomials over $\mathbb {C}$?
• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?^[53]
• The Stable Forking Conjecture for simple theories^[54]
• For which number fields does Hilbert's tenth problem hold?
• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality $\aleph _{\omega _{1}}$ does it have a model of cardinality continuum?^[55]
• Shelah's eventual Categority conjecture: For every cardinal $\lambda$ there exists a cardinal ${\displaystyle \mu (\lambda )}$ such that If an AEC K with LS(K)<= $\lambda$ is categorical in a cardinal above ${\displaystyle \mu (\lambda )}$ then it is categorical in all cardinals above ${\displaystyle \mu (\lambda )}$.^[50][56]
• Shelah's categoricity conjecture for $L_{\omega_1,\omega}$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.^[50]
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?^[57]
• If the class of atomic models of a complete first order theory is categorical in the $\aleph _{n}$, is it categorical in every cardinal?^[58][59]
• Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
• Kueker's conjecture^[60]
• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
• Lachlan's decision problem
• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
• Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
• The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?^[61]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?^[62]

Number theory

General

• Grand Riemann hypothesis
  • Generalized Riemann hypothesis
    • 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 = m² 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 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 problem
• Exponent pair conjecture
• Is π a normal number (its digits are "random")?^[63]
• Casas-Alvero conjecture
• Sato–Tate conjecture
• Find value of De Bruijn–Newman constant
• Which integers can be written as the sum of three perfect cubes?^[64]

Additive number theory

• Beal's conjecture
• Fermat–Catalan conjecture
• Goldbach's conjecture
• The values of g(k) and G(k) in Waring's problem
• Collatz conjecture (3n + 1 conjecture)
• Lander, Parkin, and Selfridge conjecture
• Diophantine quintuples
• Gilbreath's conjecture
• Erdős conjecture on arithmetic progressions
• Erdős–Turán conjecture on additive bases
• Pollock octahedral numbers conjecture
• Skolem problem
• Determine growth rate of r_k(N) (see Szemerédi's theorem)
• Minimum overlap problem

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 $e^{-1/2}$?
• Are there infinitely many repunit primes to every base except perfect power and numbers of the form −4k⁴?
• 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 square-free?
• 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 c²)?
• For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p²)?^[65]
• Can a prime p satisfy 2^p − 1 ≡ 1 (mod p²) and 3^p − 1 ≡ 1 (mod p²) simultaneously?^[66]
• Are there infinitely many Wilson primes?
• Are there infinitely many Wolstenholme primes?
• Are there any Wall–Sun–Sun primes?
• Is every Fermat number 2²ⁿ + 1 composite for $n>4$?
• Are all Fermat numbers square-free?
• 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 (so-called 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 $R(5,5)$
• The values of the Van der Waerden numbers
• Erdős–Burr conjecture

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^ℵ_ω < ℵ_ω1 (see Singular cardinals hypothesis). The best bound, ℵ_ω4, 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 ℵ_ω?
• Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
• Does the Generalized Continuum Hypothesis entail ${\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})}$ for every singular cardinal $\lambda$?
• Does the Generalized Continuum Hypothesis imply the existence of an ℵ₂-Suslin tree?

Other

• List of unsolved problems in statistics
• List of unsolved problems in computer science
• 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.^[67]
• Keating–Snaith conjecture
• Kung-Traub conjecture
• Atiyah conjecture on configurations
• Toeplitz' conjecture (open since 1911)
• Carathéodory conjecture
• Church–Turing thesis
• Weight-monodromy 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
• Erdös–Ulam problem

Problems solved since 1995

• Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)^[68]
• Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (A. Abdollahi, M. Zallaghi, 2015)^[69]
• Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)^[70]
• Erdős discrepancy problem (Terence Tao, 2015)^[71]
• Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)^[72]
• Anderson conjecture (Cheeger, Naber, 2014)^[73]
• Goldbach's weak conjecture (Harald Helfgott, 2013)^[74][75][76]
• Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)^[77][78] (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic ${\displaystyle KS_{r}}$ and ${\displaystyle KS'_{r}}$ conjectures, Bourgain-Tzafriri conjecture and ${\displaystyle R_{\epsilon }}$-conjecture)
• Virtual Haken conjecture (Agol, Groves, Manning, 2012)^[79] (and by work of Wise also virtually fibered conjecture)
• Hsiang–Lawson's conjecture (Brendle, 2012)^[80]
• Willmore conjecture (Fernando Codá Marques and André Neves, 2012)^[81]
• Ehrenpreis conjecture (Kahn, Markovic, 2011)^[82]
• Hanna Neumann conjecture (Mineyev, 2011)^[83]
• Bloch–Kato conjecture (Voevodsky, 2011)^[84] (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture^[85][86][87])
• Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)^[88]
• Density theorem (Namazi, Souto, 2010)^[89]
• Hirsch conjecture (Francisco Santos Leal, 2010)^[90][91]
• Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)^[92]
• Atiyah conjecture (Austin, 2009)^[93]
• Kauffman–Harary conjecture (Matmann, Solis, 2009)^[94]
• Surface subgroup conjecture (Kahn, Markovic, 2009)^[95]
• Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)^[96]
• Full classification of finite simple groups (Harada, Solomon, 2008)
• Geometrization conjecture (proof was completed by Morgan and Tian in 2008^[97] and it is based mostly on work of Grigori Perelman, 2002)^[98]
• Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)^{[99][100][101]}
• Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)^[102]
• Erdős–Menger conjecture (Aharoni, Berger 2007)^[103]
• Road coloring conjecture (Avraham Trahtman, 2007)^[104]
• The angel problem (Various independent proofs, 2006)^{[105][106][107][108]}
• Lax conjecture (Lewis, Parrilo, Ramana, 2005)^[109]
• The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)^[110]
• Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)^[111]
• Robertson–Seymour theorem (Robertson, Seymour, 2004)^[112]
• Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)^[113] (and also Alon–Friedgut conjecture)
• Green–Tao theorem (Ben J. Green and Terence Tao, 2004)^[114]
• Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)^[115]
• Carpenter's rule problem (Connelly, Demaine, Rote, 2003)^[116]
• Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)^[117][118]
• Milnor conjecture (Vladimir Voevodsky, 2003)^[119]
• Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)^[120]
• Nagata's conjecture (Shestakov, Umirbaev, 2003)^[121]
• Kirillov's conjecture (Baruch, 2003)^[122]
• Poincaré conjecture (Grigori Perelman, 2002)^[98]
• Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)^[123]
• Kouchnirenko’s conjecture (Haas, 2002)^[124]
• Vaught conjecture (Knight, 2002)^[125]
• Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)^[126]
• Catalan's conjecture (Preda Mihăilescu, 2002)^[127]
• n! conjecture (Haiman, 2001)^[128] (and also Macdonald positivity conjecture)
• Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)^[129]
• Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)^[130]
• Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)^[131]
• Erdős–Stewart conjecture (Florian Luca, 2001)^[132]
• Berry–Robbins problem (Atiyah, 2000)^[133]
• Erdős–Graham problem (Croot, 2000)^[134]
• Honeycomb conjecture (Thomas Hales, 1999)^[135]
• Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)^[136]
• Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)^[137][138]
• Lafforgue's theorem (Laurent Lafforgue, 1998)^[139]
• Kepler conjecture (Ferguson, Hales, 1998)^[140]
• Dodecahedral conjecture (Hales, McLaughlin, 1998)^[141]
• Ganea conjecture (Iwase, 1997)^[142]
• Torsion conjecture (Merel, 1996)^[143]
• Harary's conjecture (Chen, 1996)^[144]
• Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)^[145][146]

References

1. Eves, An Introduction to the History of Mathematics 6th Edition, Thomson, 1990, ISBN 978-0-03-029558-4.
2. Thiele, Rüdiger (2005), "On Hilbert and his twenty-four 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 0-387-25284-3
3. Guy, Richard (1994), Unsolved Problems in Number Theory (2nd ed.), Springer, p. vii, ISBN 9781489935854.
4. 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.
5. http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/dmv_091514.pdf
6. THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY
7. Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 9051994907.
8. "DARPA invests in math". CNN. 2008-10-14. Archived from the original on 2009-03-04. Retrieved 2013-01-14.
9. "Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO)". DARPA. 2007-09-10. Archived from the original on 2012-10-01. Retrieved 2013-06-25.
10. "Millennium Problems".
11. "Poincaré Conjecture". Clay Mathematics Institute. Archived from the original on 2013-12-15.
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Further reading

Books discussing recently solved problems

• Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1-84115-791-0.
• O'Shea, Donal (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9.
• Szpiro, George G. (2003). Kepler's Conjecture. Wiley. ISBN 0-471-08601-0.
• Ronan, Mark (2006). Symmetry and the Monster. Oxford. ISBN 0-19-280722-6.

Books discussing unsolved problems

• Fan Chung; Graham, Ron (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 1-56881-111-X.
• Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 0-387-20860-7.
• Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0-88385-315-9.
• Du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0-06-093558-8.
• Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0-309-08549-7.
• Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8.
• Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0-691-11748-9.
• Lizhen Ji, [various]; Yat-Sun Poon, Shing-Tung 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 1-571-46278-3.
• Waldschmidt, Michel (2004). "Open Diophantine Problems" (PDF). Moscow Mathematical Journal. 4 (1): 245–305. ISSN 1609-3321. 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 0-309-08549-7.

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 low-dimensional 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