“unsolved mathematical mysteries”✨🔭_*sketches*_🔬👨‍🔬 - ☯✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨🚀*THE 2ND NATURE*🚀

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Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved

These unsolved problems occur in multiple domains, including…

physics,

computer science,

algebra,

analysis,

combinatorics,

algebraic,

differential,

discrete and Euclidean geometries,

graph,

group,

model,

number,

set and Ramsey theories,

dynamical systems,

partial differential equations,

and more…

Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas.

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.

This article is a composite of notable unsolved problems derived from many sources, including but not limited to lists considered authoritative.

It does not claim to be comprehensive, it may not always be quite up to date, and it includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole

Lists of unsolved problems in mathematics

Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

List Number of problems Number unresolved

or incompletely resolved Proposed by Proposed in

Hilbert’s problems[2] 23 15 David Hilbert 1900

Landau’s problems[3] 4 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 14 Stephen Smale 1998

Millennium Prize problems 7 6[7] Clay Mathematics Institute 2000

Simon problems 15 <12[8][9] Barry Simon 2000

Unsolved Problems on Mathematics for the 21st Century[10] 22 – Jair Minoro Abe, Shotaro Tanaka 2001

DARPA’s math challenges[11][12] 23 – DARPA 2007

Millennium Prize Problems[edit]

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved as of July, 2020:[7]

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;[13] however, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.[14]

Unsolved problems[edit]
Algebra[edit]

Homological conjectures in commutative algebra

Finite lattice representation problem

Hilbert’s sixteenth problem

Hilbert’s fifteenth problem

Hadamard conjecture

Jacobson’s conjecture

Crouzeix’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

Rota’s basis conjecture

Uniformity conjecture

Kaplansky’s conjectures

Kummer–Vandiver conjecture

Serre’s multiplicity conjectures

Pierce–Birkhoff conjecture

Eilenberg–Ganea conjecture

Green’s conjecture

Grothendieck–Katz p-curvature conjecture

Sendov’s conjecture

Zariski–Lipman conjecture

The Dneister Notebook (Dnestrovskaya Tetrad) collects several hundred unresolved problems in algebra, particularly ring theory and modulus theory.[15]

The Erlagol Notebook (Erlagolskaya Tetrad) collects unresolved problems in algebra and model theory.[16]

Analysis[edit]

The four exponentials conjecture on the transcendence of at least one of four exponentials of combinations of irrationals[17]

Lehmer’s conjecture on the Mahler measure of non-cyclotomic polynomials[18]

The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy[19]

Schanuel’s conjecture on the transcendence degree of exponentials of linearly independent irrationals[17]

Are \gamma (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, πe, π√2, ππ, eπ2, ln π, 2e, ee, Catalan’s constant, or Khinchin’s constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?[20][21][22]

Vitushkin’s conjecture

Invariant subspace problem

Kung–Traub conjecture[23]

Regularity of solutions of Vlasov–Maxwell equations

Regularity of solutions of Euler equations

Convergence of Flint Hills series

Combinatorics[edit]

Dynamical systems[edit]

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?[30]

Birkhoff conjecture: if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?[31]

Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.

Quantum unique ergodicity conjecture[32]

Berry–Tabor conjecture

Painlevé conjecture

Games and puzzles[edit]

Combinatorial games[edit]

Sudoku:

What is the maximum number of givens for a minimal puzzle?[33]

How many puzzles have exactly one solution?[33]

How many puzzles with exactly one solution are minimal?[33]

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?[34]

What is the Turing completeness status of all unique elementary cellular automata?

Games with imperfect information[edit]

Rendezvous problem

Geometry[edit]

Algebraic geometry[edit]

Abundance conjecture

Bass conjecture

Deligne conjecture

Dixmier conjecture

Fröberg conjecture

Fujita conjecture

Hartshorne’s conjectures[35]

The Jacobian conjecture

Manin conjecture

Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory[36]

Nakai conjecture

Resolution of singularities in characteristic p

Standard conjectures on algebraic cycles

Section conjecture

Tate conjecture

Termination of flips

Virasoro conjecture

Weight-monodromy conjecture

Zariski multiplicity conjecture[37]

Covering and packing[edit]

Borsuk’s problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.

The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?[38]

The Erdős–Oler conjecture that when n is a triangular number, packing n-1 circles in an equilateral triangle requires a triangle of the same size as packing n circles[39]

The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[40]

Reinhardt’s conjecture that the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[41]

Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.

Square packing in a square: what is the asymptotic growth rate of wasted space?[42]

Ulam’s packing conjecture about the identity of the worst-packing convex solid[43]

Differential geometry[edit]

The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length[44]

The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds

The spherical Bernstein’s problem, a possible generalization of the original Bernstein’s problem

Cartan–Hadamard conjecture: Can the classical isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as Cartan–Hadamard manifolds?

Carathéodory conjecture

Chern’s conjecture (affine geometry)

Chern’s conjecture for hypersurfaces in spheres

Yau’s conjecture

Yau’s conjecture on the first eigenvalue

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.[46]

Discrete geometry[edit]

In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.

Euclidean geometry[edit]

Graph theory[edit]

Paths and cycles in graphs[edit]

Barnette’s conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle[74]

Chvátal’s toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian[75]

The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice[76]

The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs[77]

The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree[78]

The Lovász conjecture on Hamiltonian paths in symmetric graphs[79]

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.[80]

Szymanski’s conjecture

Graph coloring and labeling[edit]

An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.

Cereceda’s conjecture on the diameter of the space of colorings of degenerate graphs[81]

The Erdős–Faber–Lovász conjecture on coloring unions of cliques[82]

The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree[83]

The Hadwiger conjecture relating coloring to clique minors[84]

The Hadwiger–Nelson problem on the chromatic number of unit distance graphs[85]

Jaeger’s Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph[86]

The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index[87]

The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree[88]

Graph drawing[edit]

The Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number[89]

The Blankenship–Oporowski conjecture on the book thickness of subdivisions[90]

Conway’s thrackle conjecture[91]

Harborth’s conjecture that every planar graph can be drawn with integer edge lengths[92]

Negami’s conjecture on projective-plane embeddings of graphs with planar covers[93]

The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding[94]

Turán’s brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?[95]

Universal point sets of subquadratic size for planar graphs[96]

Word-representation of graphs[edit]

Characterise (non-)word-representable planar graphs [97][98][99][100]

Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs [101])

Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter [102]

Is the line graph of a non-word-representable graph always non-word-representable? [97][98][99][100]

Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter? [97][98][99][100]

Is it true that out of all bipartite graphs crown graphs require longest word-representants? [103]

Characterise word-representable graphs in terms of (induced) forbidden subgraphs. [97][98][99][100]

Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)? [97][98][99][100]

Miscellaneous graph theory[edit]

Group theory[edit]

The free Burnside group {\displaystyle B(2,3)} is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups {\displaystyle B(m,n)} are finite remains open.

Is every finitely presented periodic group finite?

The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?

For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?

Is every group surjunctive?

Andrews–Curtis conjecture

Herzog–Schönheim conjecture

Does generalized moonshine exist?

Are there an infinite number of Leinster groups?

Guralnick–Thompson conjecture[120]

Problems in loop theory and quasigroup theory consider generalizations of groups

The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.[121]

Model theory and formal languages[edit]

Number theory[edit]

General[edit]

6 is a perfect number because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them are odd.

Grand Riemann hypothesis

Generalized Riemann hypothesis

Riemann hypothesis

n conjecture

abc conjecture

Szpiro’s conjecture

Hilbert’s ninth problem

Hilbert’s eleventh problem

Hilbert’s twelfth problem

Carmichael’s totient function conjecture

Erdős–Straus conjecture

Erdős–Ulam problem

Pillai’s conjecture

Hall’s conjecture

Lindelöf hypothesis and its consequence, the density hypothesis for zeroes of the Riemann zeta function (see Bombieri–Vinogradov theorem)

Montgomery’s pair correlation conjecture

Hilbert–Pólya conjecture

Grimm’s conjecture

Leopoldt’s conjecture

Scholz 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

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”)?[135]

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?[136]

Erdős–Moser problem: is 11 + 21 = 31 the only solution to the Erdős–Moser equation?

Is there a covering system with odd distinct moduli?[137]

Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?[138]

The uniqueness conjecture for Markov numbers[139]

Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function[140]

Newman’s conjecture

Additive number theory[edit]

Beal’s conjecture

Fermat–Catalan conjecture

Goldbach’s conjecture

Lemoine’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 rk(N) (see Szemerédi’s theorem)

Minimum overlap problem

Do the Ulam numbers have a positive density?

Algebraic number theory[edit]

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

Greenberg’s conjectures

Hermite’s problem

Computational number theory[edit]

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

Prime numbers[edit]

Goldbach’s conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.

Goldbach conjecture

Twin prime conjecture

Polignac’s conjecture

Brocard’s Conjecture

Catalan’s Mersenne conjecture

Agoh–Giuga conjecture

Dubner’s 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

Bunyakovsky conjecture

Dickson’s conjecture

Schinzel’s hypothesis H

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 2c − 1 ≡ 1 (mod c2)?

For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[141]

Can a prime p satisfy 2p − 1 ≡ 1 (mod p2) and 3p − 1 ≡ 1 (mod p2) simultaneously?[142]

Are there infinitely many Wilson primes?

Are there infinitely many Wolstenholme primes?

Are there any Wall–Sun–Sun primes?

For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)

Is every Fermat number 22n + 1 composite for 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?

For any given integers k ≥ 1, b ≥ 2, c ≠ 0, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form (k×bn+c)/gcd(k+c,b−1) with integer n ≥ 1?

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

Set theory[edit]

Topology[edit]

Baum–Connes conjecture

Bing–Borsuk conjecture

Borel conjecture

Hilbert–Smith conjecture

Mazur’s conjectures[143]

Novikov conjecture

Telescope conjectures

Unknotting problem

Volume conjecture

Whitehead conjecture

Zeeman conjecture

Problems solved since 1995[edit]

Algebra[edit]

Connes embedding problem (Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, 2020)

Analysis[edit]

Combinatorics[edit]

Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)[146]

McMullen’s g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)[147][148]

Hirsch conjecture (Francisco Santos Leal, 2010)[149][150]

Game theory[edit]

The angel problem (Various independent proofs, 2006)[151][152][153][154]

Geometry[edit]

Yau’s conjecture (Antoine Song, 2018)[155]

Pentagonal tiling (Michaël Rao, 2017)[156]

Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)[157]

Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)[158]

Kepler conjecture (Ferguson, Hales, 1998)[159]

Dodecahedral conjecture (Hales, McLaughlin, 1998)[160]

Graph theory[edit]

Ringel’s conjecture on graceful labeling of trees (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)[161][162]

Hedetniemi’s conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)[163]

Babai’s problem (Problem 3.3 in “Spectra of Cayley graphs”) (Alireza Abdollahi, Maysam Zallaghi, 2015)[164]

Alspach’s conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)

Scheinerman’s conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)[165]

Erdős–Menger conjecture (Aharoni, Berger 2007)[166]

Road coloring conjecture (Avraham Trahtman, 2007)[167]

Group theory[edit]

Hanna Neumann conjecture (Mineyev, 2011)[168]

Density theorem (Namazi, Souto, 2010)[169]

Full classification of finite simple groups (Harada, Solomon, 2008)

Number theory[edit]

Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard, 2019)

Main conjecture in Vinogradov’s mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)[170]

Goldbach’s weak conjecture (Harald Helfgott, 2013)[171][172][173]

Serre’s modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)[174][175][176]

Fermat’s Last Theorem (Andrew Wiles and Richard Taylor, 1995)[177][178]

Ramsey theory[edit]

Burr–Erdős conjecture (Choongbum Lee, 2017)[179]

Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)[180][181]

Topology[edit]

Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020)[182] [183]

Virtual Haken conjecture (Agol, Groves, Manning, 2012)[184] (and by work of Wise also virtually fibered conjecture)

Hsiang–Lawson’s conjecture (Brendle, 2012)[185]

Ehrenpreis conjecture (Kahn, Markovic, 2011)[186]

Atiyah conjecture (Austin, 2009)[187]

Cobordism hypothesis (Jacob Lurie, 2008)[188]

Geometrization conjecture, proven by Grigori Perelman[189] in a series of preprints in 2002–2003.[190]

Spherical space form conjecture (Grigori Perelman, 2006)

Uncategorised[edit]

Erdős discrepancy problem (Terence Tao, 2015)[191]

Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)[192]

Anderson conjecture (Cheeger, Naber, 2014)[193]

Gaussian correlation inequality (Thomas Royen, 2014)[194]

Willmore conjecture (Fernando Codá Marques and André Neves, 2012)[195]

Beck’s 3-permutations conjecture (Newman, Nikolov, 2011)[196]

Bloch–Kato conjecture (Voevodsky, 2011)[197] (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture[198][199][200])

Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)[201]

Kauffman–Harary conjecture (Matmann, Solis, 2009)[202]

Surface subgroup conjecture (Kahn, Markovic, 2009)[203]

Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Lu, 2007)[204]

Nirenberg–Treves conjecture (Nils Dencker, 2005)[205][206]

Lax conjecture (Lewis, Parrilo, Ramana, 2005)[207]

The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)[208]

Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)[209]

Robertson–Seymour theorem (Robertson, Seymour, 2004)[210]

Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)[211] (and also Alon–Friedgut conjecture)

Green–Tao theorem (Ben J. Green and Terence Tao, 2004)[212]

Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)[213]

Carpenter’s rule problem (Connelly, Demaine, Rote, 2003)[214]

Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)[215][216]

Milnor conjecture (Vladimir Voevodsky, 2003)[217]

Kemnitz’s conjecture (Reiher, 2003, di Fiore, 2003)[218]

Nagata’s conjecture (Shestakov, Umirbaev, 2003)[219]

Kirillov’s conjecture (Baruch, 2003)[220]

Poincaré conjecture (Grigori Perelman, 2002)[189]

Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)[221]

Kouchnirenko’s conjecture (Haas, 2002)[222]

Vaught conjecture (Knight, 2002)[223]

Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)[224]

Catalan’s conjecture (Preda Mihăilescu, 2002)[225]

n! conjecture (Haiman, 2001)[226] (and also Macdonald positivity conjecture)

Kato’s conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)[227]

Deligne’s conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)[228]

Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)[229]

Erdős–Stewart conjecture (Florian Luca, 2001)[230]

Berry–Robbins problem (Atiyah, 2000)[231]

Erdős–Graham problem (Croot, 2000)[232]

Honeycomb conjecture (Thomas Hales, 1999)[233]

Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)[234]

Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)[235][236]

Lafforgue’s theorem (Laurent Lafforgue, 1998)[237]

Ganea conjecture (Iwase, 1997)[238]

Torsion conjecture (Merel, 1996)[239]

Harary’s conjecture (Chen, 1996)[240]

See also[edit]

List of conjectures

List of unsolved problems in statistics

List of unsolved problems in computer science

List of unsolved problems in physics

Lists of unsolved problems

Open Problems in Mathematics

The Great Mathematical Problems

Scottish Book

References[edit]

^ Eves, An Introduction to the History of Mathematics 6th Edition, Thomson, 1990, ISBN 978-0-03-029558-4.

^ Thiele, Rüdiger (2005), “On Hilbert and his twenty-four problems”, in Van Brummelen, Glen (ed.), 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 978-0-387-25284-1

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