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
Simon problems	15	Barry Simon	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 2018:^[10]

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

Combinatorics

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^[16]
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?^[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 geometry

Discrete geometry

Solving the happy ending problem for arbitrary $n$ ^[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 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?^[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 unit-length curve in the plane?^[33]
The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped 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
g-conjecture
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 5-polytopes^[36]
Covering problem of Rado
The 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?^[37]

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 $\times 2,\times 3$ 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?
Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theory
Eremenko's conjecture that every component of the escaping set of an entire transcendental function is unbounded
Is every reversible cellular automaton in three or more dimensions locally reversible?^[38]
Many problems concerning an outer billiard, for example show that outer billiards relative to almost every convex polygon has unbounded orbits.

Games and puzzles

Sudoku:
- What is the maximum number of givens for a minimal puzzle?^[39]
- How many puzzles have exactly one solution?^[39]
- How many minimal puzzles have exactly one solution?^[39]
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?^[40]

Graph theory

Paths and cycles in graphs

Barnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle^[41]
Chvátal's toughness conjecture, that there is a number $t$ such that every $t$ -tough graph is Hamiltonian^[42]
The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice^[43]
The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs^[44]
The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree^[45]
The Lovász conjecture on Hamiltonian paths in symmetric graphs^[46]
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.^[47]

Graph coloring and labeling

The Erdős–Faber–Lovász conjecture on coloring unions of cliques^[48]
The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree^[49]
The Hadwiger conjecture relating coloring to clique minors^[50]
The Hadwiger–Nelson problem on the chromatic number of unit distance graphs^[51]
Hedetniemi's conjecture on the chromatic number of tensor products of graphs^[52]
Jaeger's Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph^[53]
The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index^[54]
The Ringel–Kotzig conjecture on graceful labeling of trees^[55]
The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree^[56]

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^[57]
The Blankenship–Oporowski conjecture on the book thickness of subdivisions^[58]
Conway's thrackle conjecture^[59]
Harborth's conjecture that every planar graph can be drawn with integer edge lengths^[60]
Negami's conjecture on projective-plane embeddings of graphs with planar covers^[61]
The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding^[62]
Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?^[63]
Universal point sets of subquadratic size for planar graphs^[64]

Miscellaneous graph theory

The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph^[65]
The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs^[66]
Jørgensen's conjecture that every 6-vertex-connected K₆-minor-free graph is an apex graph^[67]
Meyniel's conjecture that cop number is $O({\sqrt n})$ ^[68]
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?^[69]
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^[70]

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 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.^[71]
Determine the structure of Keisler's order^[72]^[73]
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?^[74]
The Stable Forking Conjecture for simple theories^[75]
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?^[76]
Shelah's eventual Categority conjecture: For every cardinal $\lambda$ there exists a cardinal $\mu (\lambda )$ such that If an AEC K with LS(K)<= $\lambda$ is categorical in a cardinal above $\mu (\lambda )$ then it is categorical in all cardinals above $\mu (\lambda )$ .^[71]^[77]
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.^[71]
Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?^[78]
If the class of atomic models of a complete first order theory is categorical in the $\aleph _{n}$ , is it categorical in every cardinal?^[79]^[80]
Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
Kueker's conjecture^[81]
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?^[82]
The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?^[83]

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")?^[84]
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?^[85]
Erdős–Moser problem: is 1¹ + 2¹ = 3¹ the only solution to the Erdős–Moser equation?
Is there a covering system with odd distinct moduli?^[86]

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?

Computational number theory

Integer factorization: Can integer factorization be done in polynomial time?

Prime numbers

Brocard's Conjecture
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 $e^{-1/2}$ ?
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 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)?^[87]
Can a prime p satisfy 2^p − 1 ≡ 1 (mod p²) and 3^p − 1 ≡ 1 (mod p²) simultaneously?^[88]
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 2^2ⁿ + 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

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 ${\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})$ for every singular cardinal $\lambda$ ?
Does the Generalized Continuum Hypothesis imply the existence of an ℵ₂-Suslin tree?
Is OCA (Open coloring axiom) consistent with $2^{\aleph _{0}}>\aleph _{2}$ ?

Topology

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 group rings
Painlevé conjecture
Dixmier conjecture
Baum–Connes conjecture
Prove Turing completeness for all unique elementary cellular automata
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.^[89]
Keating–Snaith conjecture
Kung–Traub conjecture
Atiyah conjecture on configurations
Toeplitz' conjecture (open since 1911)
Carathéodory conjecture
Weight-monodromy conjecture
Berry–Tabor conjecture
Birkhoff conjecture
Guralnick–Thompson conjecture
MNOP conjecture
Mazur's conjectures
Rendezvous problem
Scholz conjecture
Nirenberg–Treves conjecture
Quantum unique ergodicity conjecture
Density hypothesis
Zhou conjecture
Erdős–Ulam problem

Problems solved since 1995

Pentagonal tiling (Michaël Rao, 2017)^[90]
Erdős–Burr conjecture (Choongbum Lee, 2017)^[91]
Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)^[92]
Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (A. Abdollahi, M. Zallaghi, 2015)^[93]
Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)^[94]
Erdős discrepancy problem (Terence Tao, 2015)^[95]
Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)^[96]
Anderson conjecture (Cheeger, Naber, 2014)^[97]
Gaussian correlation inequality (Thomas Royen, 2014)^[98]
Goldbach's weak conjecture (Harald Helfgott, 2013)^[99]^[100]^[101]
Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)^[102]^[103] (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic $KS_{r}$ and $KS'_{r}$ conjectures, Bourgain-Tzafriri conjecture and $R_{\epsilon }$ -conjecture)
Virtual Haken conjecture (Agol, Groves, Manning, 2012)^[104] (and by work of Wise also virtually fibered conjecture)
Hsiang–Lawson's conjecture (Brendle, 2012)^[105]
Willmore conjecture (Fernando Codá Marques and André Neves, 2012)^[106]
Ehrenpreis conjecture (Kahn, Markovic, 2011)^[107]
Hanna Neumann conjecture (Mineyev, 2011)^[108]
Bloch–Kato conjecture (Voevodsky, 2011)^[109] (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture^[110]^[111]^[112])
Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)^[113]
Density theorem (Namazi, Souto, 2010)^[114]
Hirsch conjecture (Francisco Santos Leal, 2010)^[115]^[116]
Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)^[117]
Atiyah conjecture (Austin, 2009)^[118]
Kauffman–Harary conjecture (Matmann, Solis, 2009)^[119]
Surface subgroup conjecture (Kahn, Markovic, 2009)^[120]
Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)^[121]
Cobordism hypothesis (Jacob Lurie, 2008)^[122]
Full classification of finite simple groups (Harada, Solomon, 2008)
Geometrization conjecture (proof was completed by Morgan and Tian in 2008^[123] and it is based mostly on work of Grigori Perelman, 2002)^[124]
Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)^[125]^[126]^[127]
Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)^[128]
Erdős–Menger conjecture (Aharoni, Berger 2007)^[129]
Road coloring conjecture (Avraham Trahtman, 2007)^[130]
The angel problem (Various independent proofs, 2006)^[131]^[132]^[133]^[134]
Lax conjecture (Lewis, Parrilo, Ramana, 2005)^[135]
The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)^[136]
Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)^[137]
Robertson–Seymour theorem (Robertson, Seymour, 2004)^[138]
Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)^[139] (and also Alon–Friedgut conjecture)
Green–Tao theorem (Ben J. Green and Terence Tao, 2004)^[140]
Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)^[141]
Carpenter's rule problem (Connelly, Demaine, Rote, 2003)^[142]
Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)^[143]^[144]
Milnor conjecture (Vladimir Voevodsky, 2003)^[145]
Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)^[146]
Nagata's conjecture (Shestakov, Umirbaev, 2003)^[147]
Kirillov's conjecture (Baruch, 2003)^[148]
Poincaré conjecture (Grigori Perelman, 2002)^[124]
Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)^[149]
Kouchnirenko’s conjecture (Haas, 2002)^[150]
Vaught conjecture (Knight, 2002)^[151]
Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)^[152]
Catalan's conjecture (Preda Mihăilescu, 2002)^[153]
n! conjecture (Haiman, 2001)^[154] (and also Macdonald positivity conjecture)
Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)^[155]
Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)^[156]
Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)^[157]
Erdős–Stewart conjecture (Florian Luca, 2001)^[158]
Berry–Robbins problem (Atiyah, 2000)^[159]
Erdős–Graham problem (Croot, 2000)^[160]
Honeycomb conjecture (Thomas Hales, 1999)^[161]
Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)^[162]
Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)^[163]^[164]
Lafforgue's theorem (Laurent Lafforgue, 1998)^[165]
Kepler conjecture (Ferguson, Hales, 1998)^[166]
Dodecahedral conjecture (Hales, McLaughlin, 1998)^[167]
Ganea conjecture (Iwase, 1997)^[168]
Torsion conjecture (Merel, 1996)^[169]
Harary's conjecture (Chen, 1996)^[170]
Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)^[171]^[172]

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↑ 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 , accessed 15 December 2014.
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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
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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
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Aizenman, Michael. "Open Problems in Mathematical Physics".
15 Problems in Mathematical Physics

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