List of unsolved problems in mathematics
Many mathematical problems have not been solved yet. These unsolved problems occur in multiple domains, including theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, and partial differential equations. 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 mathematicsEdit
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||23||15||David Hilbert||1900|
|Landau's problems||4||4||Edmund Landau||1912|
|Taniyama's problems||36||-||Yutaka Taniyama||1955|
|Thurston's 24 questions||24||-||William Thurston||1982|
|Smale's problems||18||14||Stephen Smale||1998|
|Millennium Prize problems||7||6||Clay Mathematics Institute||2000|
|Simon problems||15||<12||Barry Simon||2000|
|Unsolved Problems on Mathematics for the 21st Century||22||-||Jair Minoro Abe, Shotaro Tanaka||2001|
|DARPA's math challenges||23||-||DARPA||2007|
Millennium Prize ProblemsEdit
- Birch and Swinnerton-Dyer conjecture
- Hodge conjecture
- Navier–Stokes existence and smoothness
- P versus NP
- Riemann hypothesis
- Yang–Mills existence and mass gap
The seventh problem, the Poincaré conjecture, has been solved; 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.
- The Dneister Notebook (Dnestrovskaya Tetrad) collects several hundred unresolved problems in algebra, particularly ring theory and modulus theory.
- The Erlagol Notebook (Erlagolskaya Tetrad) collects unresolved problems in algebra and model theory.
Conjectures and problemsEdit
- Birch–Tate conjecture
- Bombieri–Lang conjecture
- Crouzeix's conjecture
- Demazure conjecture
- Eilenberg–Ganea conjecture
- Farrell–Jones conjecture
- Bost conjecture
- Finite lattice representation problem
- Green's conjecture
- Grothendieck–Katz p-curvature conjecture
- Hadamard conjecture
- Hilbert's fifteenth problem
- Hilbert's sixteenth problem
- Homological conjectures in commutative algebra
- Jacobson's conjecture
- Kaplansky's conjectures
- Köthe conjecture
- Kummer–Vandiver conjecture
- Existence of perfect cuboids and associated cuboid conjectures
- Pierce–Birkhoff conjecture
- Rota's basis conjecture
- Sendov's conjecture
- Serre's conjecture II
- Serre's multiplicity conjectures
- Uniform boundedness conjecture for rational points
- Wild problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
- Zariski–Lipman conjecture
- Zauner's conjecture: existence of SIC-POVMs in all dimensions
Conjectures and problemsEdit
- Brennan conjecture
- The four exponentials conjecture on the transcendence of at least one of four exponentials of combinations of irrationals
- Invariant subspace problem
- Kung–Traub conjecture
- Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials
- The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy
- Schanuel's conjecture on the transcendence degree of exponentials of linearly independent irrationals
- Vitushkin's conjecture
- Are (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, πe, π√, 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?
- Regularity of solutions of Euler equations
- Convergence of Flint Hills series
- Regularity of solutions of Vlasov–Maxwell equations
Conjectures and problemsEdit
- 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?
- Problems in Latin squares - Open questions concerning Latin squares
- 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?
- No-three-in-line problem
- 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 values of the Dedekind numbers for .
- Give a combinatorial interpretation of the Kronecker coefficients.
- The values of the Ramsey numbers, particularly
- Finding a function to model n-step self-avoiding walks.
- The values of the Van der Waerden numbers
Conjectures and problemsEdit
- Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theory
- Quantum chaos: Berry–Tabor conjecture
- Birkhoff conjecture: if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?
- Collatz conjecture (3n + 1 conjecture)
- Eremenko's conjecture that every component of the escaping set of an entire transcendental function is unbounded
- Furstenberg conjecture – Is every invariant and ergodic measure for the action on the circle either Lebesgue or atomic?
- Kaplan–Yorke conjecture
- Margulis conjecture – Measure classification for diagonalizable actions in higher-rank groups
- MLC conjecture – Is the Mandelbrot set locally connected?
- Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.
- Painlevé conjecture
- Quantum unique ergodicity conjecture
- 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?
- Lyapunov function: 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?
- Is every reversible cellular automaton in three or more dimensions locally reversible?
Games and puzzlesEdit
- 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?
- What is the Turing completeness status of all unique elementary cellular automata?
Games with imperfect informationEdit
- Abundance conjecture
- Bass conjecture
- Deligne conjecture
- Dixmier conjecture
- Fröberg conjecture
- Fujita conjecture
- Hartshorne's conjectures
- The Jacobian conjecture
- Manin conjecture
- Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory
- Nakai conjecture
- Parshin's conjecture
- Section conjecture
- Standard conjectures on algebraic cycles
- Tate conjecture
- Virasoro conjecture
- Weight-monodromy conjecture
- Zariski multiplicity conjecture
Covering and packingEdit
Conjectures and problemsEdit
- 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?
- The Erdős–Oler conjecture that when is a triangular number, packing circles in an equilateral triangle requires a triangle of the same size as packing circles
- The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24
- Reinhardt's conjecture that the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets
- 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?
- Ulam's packing conjecture about the identity of the worst-packing convex solid
Conjectures and problemsEdit
- The spherical Bernstein's problem, a possible generalization of the original Bernstein's problem
- Carathéodory conjecture
- 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?
- Chern's conjecture (affine geometry)
- Chern's conjecture for hypersurfaces in spheres
- 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.
- 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
- The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds
- Yau's conjecture
- Yau's conjecture on the first eigenvalue
Conjectures and problemsEdit
- The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies
- Solving the happy ending problem for arbitrary 
- Improving lower and upper bounds for the Heilbronn triangle problem.
- Kalai's 3d conjecture on the least possible number of faces of centrally symmetric polytopes.
- The Kobon triangle problem on triangles in line arrangements
- The Kusner conjecture that at most points can be equidistant in spaces
- The McMullen problem on projectively transforming sets of points into convex position
- Opaque forest problem
Conjectures and problemsEdit
- The Atiyah conjecture on configurations
- Bellman's lost in a forest problem – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation
- Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?
- Ehrhart's volume conjecture
- The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?
- Falconer's conjecture that sets of Hausdorff dimension greater than in must have a distance set of nonzero Lebesgue measure
- Inscribed square problem, also known as Toeplitz' conjecture – does every Jordan curve have an inscribed square?
- The Kakeya conjecture – do -dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to ?
- The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem
- Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one
- Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar.
- Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?
- The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
- Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edge-unfolding?
- The Thomson problem – what is the minimum energy configuration of mutually-repelling particles on a unit sphere?
- Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?
- Dissection into orthoschemes – is it possible for simplices of every dimension?
Graph coloring and labelingEdit
Conjectures and problemsEdit
- Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs
- The Erdős–Faber–Lovász conjecture on coloring unions of cliques
- The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree
- The Hadwiger conjecture relating coloring to clique minors
- The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
- Jaeger's Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph
- The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index
- The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree
Conjectures and problemsEdit
- The Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number
- The Blankenship–Oporowski conjecture on the book thickness of subdivisions
- Conway's thrackle conjecture
- Harborth's conjecture that every planar graph can be drawn with integer edge lengths
- Negami's conjecture on projective-plane embeddings of graphs with planar covers
- The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding
- Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?
Paths and cycles in graphsEdit
Conjectures and problemsEdit
- Barnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle
- Chvátal's toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian
- The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice
- The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
- The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree
- The Lovász conjecture on Hamiltonian paths in symmetric graphs
- 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.
- Szymanski's conjecture
Word-representation of graphsEdit
- Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter?
- Characterise (non-)word-representable planar graphs
- Characterise word-representable graphs in terms of (induced) forbidden subgraphs.
- Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs)
- 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
- Is it true that out of all bipartite graphs, crown graphs require longest word-representants?
- Is the line graph of a non-word-representable graph always non-word-representable?
- Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?
Miscellaneous graph theoryEdit
Conjectures and problemsEdit
- Brouwer's conjecture
- Conway's 99-graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?
- The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph
- The GNRS conjecture on whether minor-closed graph families have embeddings with bounded distortion
- Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs
- The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs
- Jørgensen's conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph
- Meyniel's conjecture that cop number is 
- The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.
- The second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?
- Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?
- Sumner's conjecture: does every -vertex tournament contain as a subgraph every -vertex oriented tree?
- 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
- Zarankiewicz problem
- Does a Moore graph with girth 5 and degree 57 exist?
- What is the largest possible pathwidth of an n-vertex cubic graph?
- The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.
Conjectures and problemsEdit
- Andrews–Curtis conjecture
- Guralnick–Thompson conjecture
- Herzog–Schönheim conjecture
- The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
- Problems in loop theory and quasigroup theory consider generalizations of groups
- Are there an infinite number of Leinster groups?
- Does generalized moonshine exist?
- For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
- Is every finitely presented periodic group finite?
- Is every group surjunctive?
Model theory and formal languagesEdit
Conjectures and problemsEdit
- The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in is a simple algebraic group over an algebraically closed field.
- Generalized star height problem
- For which number fields does Hilbert's tenth problem hold?
- Kueker's conjecture
- The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for -saturated models of a countable theory.
- 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.
- Shelah's eventual categoricity 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 .
- The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
- The Stable Forking Conjecture for simple theories
- Tarski's exponential function problem
- 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?
- The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?
- Vaught's conjecture
- 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?
- Do the Henson graphs have the finite model property?
- Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
- Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
- If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?
- Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
- (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?
- Is the theory of the field of Laurent series over decidable? of the field of polynomials over ?
- Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?
Conjectures, problems and hypothesesEdit
- André–Oort conjecture
- Beilinson conjecture
- Brocard's problem: existence of integers, (n,m), such that n! + 1 = m2 other than n = 4, 5, 7
- Carmichael's totient function conjecture
- Casas-Alvero conjecture
- Catalan–Dickson conjecture on aliquot sequences
- Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
- Erdős–Moser problem: is 11 + 21 = 31 the only solution to the Erdős–Moser equation?
- Erdős–Straus conjecture
- Erdős–Ulam problem
- Exponent pair conjecture (Van der Corput's method)
- 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?
- Goormaghtigh conjecture
- Grand Riemann hypothesis
- Grimm's conjecture
- Hall's conjecture
- Hardy–Littlewood zeta-function conjectures
- Hilbert's eleventh problem
- Hilbert's ninth problem
- Hilbert's twelfth problem
- Hilbert–Pólya conjecture
- Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function
- Lehmer's totient problem: if φ(n) divides n − 1, must n be prime?
- Leopoldt's conjecture
- Lindelöf hypothesis and its consequence, the density hypothesis for zeroes of the Riemann zeta function (see Bombieri–Vinogradov theorem)
- Littlewood conjecture
- Mahler's 3/2 problem
- Montgomery's pair correlation conjecture
- n conjecture
- Newman's conjecture
- Pillai's conjecture
- Piltz divisor problem, especially Dirichlet's divisor problem
- Ramanujan–Petersson conjecture
- Sato–Tate conjecture
- Scholz conjecture
- Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
- The uniqueness conjecture for Markov numbers
- Vojta's conjecture
- Are there any pairs of amicable numbers which have opposite parity?
- Are there any pairs of betrothed numbers which have same parity?
- Are there any pairs of relatively prime amicable numbers?
- Are there infinitely many amicable numbers?
- Are there infinitely many betrothed numbers?
- Are there infinitely many perfect numbers?
- Do any Lychrel numbers exist?
- Do any odd perfect numbers exist?
- Do any odd weird numbers exist?
- Do any Taxicab(5, 2, n) exist for n > 1?
- Do quasiperfect numbers exist?
- Is there a covering system with odd distinct moduli?
- Is π a normal number (its digits are "random")?
- Is 10 a solitary number?
- Which integers can be written as the sum of three perfect cubes?
- Find the value of the De Bruijn–Newman constant
Additive number theoryEdit
Conjectures and problemsEdit
- Beal's conjecture
- Erdős conjecture on arithmetic progressions
- Erdős–Turán conjecture on additive bases
- Fermat–Catalan conjecture
- Gilbreath's conjecture
- Goldbach's conjecture
- Lander, Parkin, and Selfridge conjecture
- Lemoine's conjecture
- Minimum overlap problem
- Pollock's conjectures
- Skolem problem
- The values of g(k) and G(k) in Waring's problem
- Do the Ulam numbers have a positive density?
- Determine growth rate of rk(N) (see Szemerédi's theorem)
Algebraic number theoryEdit
Conjectures and problemsEdit
- Class number problem: are there infinitely many real quadratic number fields with unique factorization?
- Fontaine–Mazur conjecture
- Gan–Gross–Prasad conjecture
- Greenberg's conjectures
- Hermite's problem
- Kummer–Vandiver conjecture
- Selberg's 1/4 conjecture
- Stark conjectures (including Brumer–Stark conjecture)
- Characterize all algebraic number fields that have some power basis.
Computational number theoryEdit
- Integer factorization: Can integer factorization be done in polynomial time?
Conjectures, problems and hypothesesEdit
- Agoh–Giuga conjecture
- Artin's conjecture on primitive roots
- Brocard's conjecture
- Bunyakovsky conjecture
- Catalan's Mersenne conjecture
- Dickson's conjecture
- Dubner's conjecture
- Elliott–Halberstam conjecture
- Erdős–Mollin–Walsh conjecture
- Feit–Thompson conjecture
- Fortune's conjecture (that no Fortunate number is composite)
- 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?
- Gillies' conjecture
- Goldbach conjecture
- Landau's problems
- Problems associated to Linnik's theorem
- New Mersenne conjecture
- Polignac's conjecture
- Schinzel's hypothesis H
- Is 78,557 the lowest Sierpiński number (so-called Selfridge's conjecture)?
- Twin prime conjecture
- Does the conjectural converse of Wolstenholme's theorem hold for all natural numbers?
- Are all Fermat numbers square-free?
- Are all Mersenne numbers of prime index square-free?
- Are there any composite c satisfying 2c − 1 ≡ 1 (mod c2)?
- Are there any Wall–Sun–Sun primes?
- Are there any Wieferich primes in base 47?
- Are there infinitely many Carol primes?
- Are there infinitely many cousin primes?
- Are there infinitely many Cullen primes?
- Are there infinitely many Fibonacci primes?
- Are there infinitely many Kynea primes?
- Are there infinitely many Lucas primes?
- Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
- Are there infinitely many Newman–Shanks–Williams primes?
- Are there infinitely many palindromic primes to every base?
- Are there infinitely many Pell primes?
- Are there infinitely many Pierpont primes?
- Are there infinitely many prime quadruplets?
- Are there infinitely many regular primes, and if so is their relative density ?
- Are there infinitely many sexy primes?
- Are there infinitely many Sophie Germain primes?
- Are there infinitely many Wagstaff primes?
- Are there infinitely many Wieferich primes?
- Are there infinitely many Wilson primes?
- Are there infinitely many Wolstenholme primes?
- Are there infinitely many Woodall primes?
- Can a prime p satisfy 2p − 1 ≡ 1 (mod p2) and 3p − 1 ≡ 1 (mod p2) simultaneously?
- Does every prime number appear in the Euclid–Mullin sequence?
- Find the smallest Skewes' number
- 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)
- For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?
- 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?
- 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?
- 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?
- Is every Fermat number 22n + 1 composite for ?
- Is 509,203 the lowest Riesel number?
Conjectures, problems, and hypothesesEdit
- (Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?
- Does the Generalized Continuum Hypothesis entail for every singular cardinal ?
- Does the Generalized Continuum Hypothesis imply the existence of an ℵ2-Suslin tree?
- If ℵω is a strong limit cardinal, then 2ℵω < ℵω1 (see Singular cardinals hypothesis). The best bound, ℵω4, was obtained by Shelah using his PCF theory.
- The problem of finding the ultimate core model, one that contains all large cardinals.
- Woodin's Ω-conjecture
- Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
- Does there exist a Jónsson algebra on ℵω?
- Is OCA (Open coloring axiom) consistent with ?
- Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
Conjectures and problemsEdit
Problems solved since 1995Edit
- Connes embedding problem (Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, 2020)
- Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013) (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic and conjectures, Bourgain-Tzafriri conjecture and -conjecture)
- Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)
- 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)
- Hirsch conjecture (Francisco Santos Leal, 2010)
- Yau's conjecture (Antoine Song, 2018)
- Pentagonal tiling (Michaël Rao, 2017)
- Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)
- Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)
- Kepler conjecture (Ferguson, Hales, 1998)
- Dodecahedral conjecture (Hales, McLaughlin, 1998)
- Ringel's conjecture on graceful labeling of trees (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)
- Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)
- Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (Alireza Abdollahi, Maysam Zallaghi, 2015)
- Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
- Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)
- Erdős–Menger conjecture (Aharoni, Berger 2007)
- Road coloring conjecture (Avraham Trahtman, 2007)
- Hanna Neumann conjecture (Mineyev, 2011)
- Density theorem (Namazi, Souto, 2010)
- Full classification of finite simple groups (Harada, Solomon, 2008)
- Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard, 2019)
- Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)
- Goldbach's weak conjecture (Harald Helfgott, 2013)
- Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)
- Burr–Erdős conjecture (Choongbum Lee, 2017)
- Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)
Theoretical computer scienceEdit
- Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020)
- Virtual Haken conjecture (Agol, Groves, Manning, 2012) (and by work of Wise also virtually fibered conjecture)
- Hsiang–Lawson's conjecture (Brendle, 2012)
- Ehrenpreis conjecture (Kahn, Markovic, 2011)
- Atiyah conjecture (Austin, 2009)
- Cobordism hypothesis (Jacob Lurie, 2008)
- Geometrization conjecture, proven by Grigori Perelman in a series of preprints in 2002–2003.
- Spherical space form conjecture (Grigori Perelman, 2006)
- Erdős discrepancy problem (Terence Tao, 2015)
- Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)
- Anderson conjecture (Cheeger, Naber, 2014)
- Gaussian correlation inequality (Thomas Royen, 2014)
- Willmore conjecture (Fernando Codá Marques and André Neves, 2012)
- Beck's 3-permutations conjecture (Newman, Nikolov, 2011)
- Bloch–Kato conjecture (Voevodsky, 2011) (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture)
- Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)
- Kauffman–Harary conjecture (Matmann, Solis, 2009)
- Surface subgroup conjecture (Kahn, Markovic, 2009)
- Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Lu, 2007)
- Nirenberg–Treves conjecture (Nils Dencker, 2005)
- Lax conjecture (Lewis, Parrilo, Ramana, 2005)
- The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)
- Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)
- Robertson–Seymour theorem (Robertson, Seymour, 2004)
- Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004) (and also Alon–Friedgut conjecture)
- Green–Tao theorem (Ben J. Green and Terence Tao, 2004)
- Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)
- Carpenter's rule problem (Connelly, Demaine, Rote, 2003)
- Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)
- Milnor conjecture (Vladimir Voevodsky, 2003)
- Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)
- Nagata's conjecture (Shestakov, Umirbaev, 2003)
- Kirillov's conjecture (Baruch, 2003)
- Poincaré conjecture (Grigori Perelman, 2002)
- Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)
- Kouchnirenko’s conjecture (Haas, 2002)
- Vaught conjecture (Knight, 2002)
- Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)
- Catalan's conjecture (Preda Mihăilescu, 2002)
- n! conjecture (Haiman, 2001) (and also Macdonald positivity conjecture)
- Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)
- Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)
- Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)
- Erdős–Stewart conjecture (Florian Luca, 2001)
- Berry–Robbins problem (Atiyah, 2000)
- Erdős–Graham problem (Croot, 2000)
- Honeycomb conjecture (Thomas Hales, 1999)
- Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)
- Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)
- Lafforgue's theorem (Laurent Lafforgue, 1998)
- Ganea conjecture (Iwase, 1997)
- Torsion conjecture (Merel, 1996)
- Harary's conjecture (Chen, 1996)
- 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
- Guy, Richard (1994), Unsolved Problems in Number Theory (2nd ed.), Springer, p. vii, ISBN 978-1-4899-3585-4, archived from the original on 2019-03-23, retrieved 2016-09-22.
- 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.
- "Archived copy" (PDF). Archived from the original (PDF) on 2016-02-08. Retrieved 2016-01-22.CS1 maint: archived copy as title (link)
- "THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY" (PDF). Archived (PDF) from the original on 2016-04-10. Retrieved 2016-02-09.
- "Millennium Problems". Archived from the original on 2017-06-06. Retrieved 2015-01-20.
- "Fields Medal awarded to Artur Avila". Centre national de la recherche scientifique. 2014-08-13. Archived from the original on 2018-07-10. Retrieved 2018-07-07.
- Bellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Archived from the original on 2016-10-21. Retrieved 2018-07-07.
- Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 978-9051994902.
- "DARPA invests in math". CNN. 2008-10-14. Archived from the original on 2009-03-04. Retrieved 2013-01-14.
- "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.
- "Poincaré Conjecture". Clay Mathematics Institute. Archived from the original on 2013-12-15.
- "Smooth 4-dimensional Poincare conjecture". Archived from the original on 2018-01-25. Retrieved 2019-08-06.
- Dnestrovskaya notebook (PDF) (in Russian), The Russian Academy of Sciences, 1993
"Dneister Notebook: Unsolved Problems in the Theory of Rings and Modules" (PDF), University of Saskatchewan, retrieved 2019-08-15
- Erlagol notebook (PDF) (in Russian), The Novosibirsk State University, 2018
- Dowling, T A (February 1973). "A class of geometric lattices based on finite groups". Department of Statistics, University of North Carolina, Chapel Hill, NC 27514, USA & the Department of Mathematics, Ohio State University, Columbus, Ohio 43210. Journal of Combinatorial Theory, Series B. Institute of Statistics Mimeo Series No. 825. 14 (1): 61–86. doi:10.1016/S0095-8956(73)80007-3. Retrieved 18 June 2021 – via Google scholar.
Copyright © 2021 Elsevier B.V. or its licensors or contributors. https://www.elsevier.com/about/policies/open-access-licenses/elsevier-user-license, "A class of geometric lattices based on finite groups" was supported in part by the Air Force Office of Scientific Research under Contract AFOSR-68-1415.
- Waldschmidt, Michel (2013), Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Springer, pp. 14, 16, ISBN 9783662115695
- Kung, H. T.; Traub, Joseph Frederick (1974), "Optimal order of one-point and multipoint iteration", Journal of the ACM, 21 (4): 643–651, doi:10.1145/321850.321860, S2CID 74921
- Smyth, Chris (2008), "The Mahler measure of algebraic numbers: a survey", in McKee, James; Smyth, Chris (eds.), Number Theory and Polynomials, London Mathematical Society Lecture Note Series, 352, Cambridge University Press, pp. 322–349, ISBN 978-0-521-71467-9
- Berenstein, Carlos A. (2001) , "Pompeiu problem", Encyclopedia of Mathematics, EMS Press
- For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi ( Archived 2014-12-06 at the Wayback Machine), e ( Archived 2014-11-21 at the Wayback Machine), Khinchin's Constant ( Archived 2014-11-05 at the Wayback Machine), irrational numbers ( Archived 2015-03-27 at the Wayback Machine), transcendental numbers ( Archived 2014-11-13 at the Wayback Machine), and irrationality measures ( Archived 2015-04-21 at the Wayback Machine) 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  Archived 2014-12-16 at the Wayback Machine, 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  Archived 2014-01-17 at the Wayback Machine, accessed 15 December 2014.
- Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, CiteSeerX 10.1.1.38.7841, doi:10.1007/BF01110378, MR 1368815, S2CID 14793475.
- Tao, Terence (2017), "Some remarks on the lonely runner conjecture", arXiv:1701.02048 [math.CO]
- Bruhn, Henning; Schaudt, Oliver (2015), "The journey of the union-closed sets conjecture" (PDF), Graphs and Combinatorics, 31 (6): 2043–2074, arXiv:1309.3297, doi:10.1007/s00373-014-1515-0, MR 3417215, S2CID 17531822, archived (PDF) from the original on 2017-08-08, retrieved 2017-07-18
- Dedekind Numbers and Related Sequences
- 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, JSTOR 2371542, MR 1507301, PMC 1076971, PMID 16577800
- Liśkiewicz, Maciej; Ogihara, Mitsunori; Toda, Seinosuke (2003-07-28). "The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes". Theoretical Computer Science. 304 (1): 129–156. doi:10.1016/S0304-3975(03)00080-X.
- Kaloshin, Vadim; Sorrentino, Alfonso (2018). "On the local Birkhoff conjecture for convex billiards". Annals of Mathematics. 188 (1): 315–380. arXiv:1612.09194. doi:10.4007/annals.2018.188.1.6. S2CID 119171182.
- Sarnak, Peter (2011), "Recent progress on the quantum unique ergodicity conjecture", Bulletin of the American Mathematical Society, 48 (2): 211–228, doi:10.1090/S0273-0979-2011-01323-4, MR 2774090
- 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, Bibcode:2009LNCS.5715....6K, doi:10.1007/978-3-642-03745-0_5, ISBN 978-3-642-03744-3
- http://english.log-it-ex.com Archived 2017-11-10 at the Wayback Machine Ten open questions about Sudoku (2012-01-21).
- "Higher-Dimensional Tic-Tac-Toe". PBS Infinite Series. YouTube. 2017-09-21. Archived from the original on 2017-10-11. Retrieved 2018-07-29.
- Barlet, Daniel; Peternell, Thomas; Schneider, Michael (1990). "On two conjectures of Hartshorne's". Mathematische Annalen. 286 (1–3): 13–25. doi:10.1007/BF01453563. S2CID 122151259.
- Maulik, Davesh; Nekrasov, Nikita; Okounov, Andrei; Pandharipande, Rahul (2004-06-05), Gromov–Witten theory and Donaldson–Thomas theory, I, arXiv:math/0312059, Bibcode:2003math.....12059M
- Zariski, Oscar (1971). "Some open questions in the theory of singularities". Bulletin of the American Mathematical Society. 77 (4): 481–491. doi:10.1090/S0002-9904-1971-12729-5. MR 0277533.
- Bereg, Sergey; Dumitrescu, Adrian; Jiang, Minghui (2010), "On covering problems of Rado", Algorithmica, 57 (3): 538–561, doi:10.1007/s00453-009-9298-z, MR 2609053, S2CID 6511998
- Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, JSTOR 2324212, MR 1252928
- Conway, John H.; Neil J.A. Sloane (1999), Sphere Packings, Lattices and Groups (3rd ed.), New York: Springer-Verlag, pp. 21–22, ISBN 978-0-387-98585-5
- Hales, Thomas (2017), The Reinhardt conjecture as an optimal control problem, arXiv:1703.01352
- Brass, Peter; Moser, William; Pach, János (2005), Research Problems in Discrete Geometry, New York: Springer, p. 45, ISBN 978-0387-23815-9, MR 2163782
- Gardner, Martin (1995), New Mathematical Diversions (Revised Edition), Washington: Mathematical Association of America, p. 251
- Barros, Manuel (1997), "General Helices and a Theorem of Lancret", Proceedings of the American Mathematical Society, 125 (5): 1503–1509, doi:10.1090/S0002-9939-97-03692-7, JSTOR 2162098
- Katz, Mikhail G. (2007), Systolic geometry and topology, Mathematical Surveys and Monographs, 137, American Mathematical Society, Providence, RI, p. 57, doi:10.1090/surv/137, ISBN 978-0-8218-4177-8, MR 2292367
- Rosenberg, Steven (1997), The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds, London Mathematical Society Student Texts, 31, Cambridge: Cambridge University Press, pp. 62–63, doi:10.1017/CBO9780511623783, ISBN 978-0-521-46300-3, MR 1462892
- Boltjansky, V.; Gohberg, I. (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
- Morris, Walter D.; Soltan, Valeriu (2000), "The Erdős-Szekeres problem on points in convex position—a survey", Bull. Amer. Math. Soc., 37 (4): 437–458, doi:10.1090/S0273-0979-00-00877-6, MR 1779413; Suk, Andrew (2016), "On the Erdős–Szekeres convex polygon problem", J. Amer. Math. Soc., 30 (4): 1047–1053, arXiv:1604.08657, doi:10.1090/jams/869, S2CID 15732134
- Kalai, Gil (1989), "The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics, 5 (1): 389–391, doi:10.1007/BF01788696, MR 1554357, S2CID 8917264.
- Weisstein, Eric W. "Kobon Triangle". MathWorld.
- Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", American Mathematical Monthly, 90 (3): 196–200, doi:10.2307/2975549, JSTOR 2975549, MR 1540158
- Matoušek, Jiří (2002), Lectures on discrete geometry, Graduate Texts in Mathematics, 212, Springer-Verlag, New York, p. 206, doi:10.1007/978-1-4613-0039-7, ISBN 978-0-387-95373-1, MR 1899299
- 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 978-0-387-23815-9, MR 2163782
- Dey, Tamal K. (1998), "Improved bounds for planar k-sets and related problems", Discrete Comput. Geom., 19 (3): 373–382, doi:10.1007/PL00009354, MR 1608878; Tóth, Gábor (2001), "Point sets with many k-sets", Discrete Comput. Geom., 26 (2): 187–194, doi:10.1007/s004540010022, MR 1843435.
- Aronov, Boris; Dujmović, Vida; Morin, Pat; Ooms, Aurélien; Schultz Xavier da Silveira, Luís Fernando (2019), "More Turán-type theorems for triangles in convex point sets", Electronic Journal of Combinatorics, 26 (1): P1.8, arXiv:1706.10193, Bibcode:2017arXiv170610193A, doi:10.37236/7224, archived from the original on 2019-02-18, retrieved 2019-02-18
- Atiyah, Michael (2001), "Configurations of points", Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 359 (1784): 1375–1387, Bibcode:2001RSPTA.359.1375A, doi:10.1098/rsta.2001.0840, ISSN 1364-503X, MR 1853626, S2CID 55833332
- Finch, S. R.; Wetzel, J. E. (2004), "Lost in a forest", American Mathematical Monthly, 11 (8): 645–654, doi:10.2307/4145038, JSTOR 4145038, MR 2091541
- Solomon, Yaar; Weiss, Barak (2016), "Dense forests and Danzer sets", Annales Scientifiques de l'École Normale Supérieure, 49 (5): 1053–1074, arXiv:1406.3807, doi:10.24033/asens.2303, MR 3581810, S2CID 672315; Conway, John H., Five $1,000 Problems (Update 2017) (PDF), On-Line Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12
- 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/s00283-011-9255-y, MR 2902144, S2CID 10747746
- Arutyunyants, G.; Iosevich, A. (2004), "Falconer conjecture, spherical averages and discrete analogs", in Pach, János (ed.), Towards a Theory of Geometric Graphs, Contemp. Math., 342, Amer. Math. Soc., Providence, RI, pp. 15–24, doi:10.1090/conm/342/06127, ISBN 9780821834848, MR 2065249
- Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100
- Katz, Nets; Tao, Terence (2002), "Recent progress on the Kakeya conjecture", Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), Publicacions Matemàtiques (Vol. Extra): 161–179, CiteSeerX 10.1.1.241.5335, doi:10.5565/PUBLMAT_Esco02_07, MR 1964819, S2CID 77088
- Weaire, Denis, ed. (1997), The Kelvin Problem, CRC Press, p. 1, ISBN 9780748406326
- Brass, Peter; Moser, William; Pach, János (2005), Research problems in discrete geometry, New York: Springer, p. 457, ISBN 9780387299297, MR 2163782
- Mahler, Kurt (1939). "Ein Minimalproblem für konvexe Polygone". Mathematica (Zutphen) B: 118–127.
- 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, archived (PDF) from the original on 2015-04-20, retrieved 2014-05-14
- 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
- Ghomi, Mohammad (2018-01-01). "D "urer's Unfolding Problem for Convex Polyhedra". Notices of the American Mathematical Society. 65 (1): 25–27. doi:10.1090/noti1609. ISSN 0002-9920.
- Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly, 59 (9): 606–611, doi:10.2307/2306764, JSTOR 2306764, MR 0050303
- Howards, Hugh Nelson (2013), "Forming the Borromean rings out of arbitrary polygonal unknots", Journal of Knot Theory and Its Ramifications, 22 (14): 1350083, 15, arXiv:1406.3370, doi:10.1142/S0218216513500831, MR 3190121, S2CID 119674622
- Brandts, Jan; Korotov, Sergey; Křížek, Michal; Šolc, Jakub (2009), "On nonobtuse simplicial partitions" (PDF), SIAM Review, 51 (2): 317–335, Bibcode:2009SIAMR..51..317B, doi:10.1137/060669073, MR 2505583, archived (PDF) from the original on 2018-11-04, retrieved 2018-11-22. See in particular Conjecture 23, p. 327.
- ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, archived from the original on October 5, 2016, retrieved 2016-10-04.
- Bousquet, Nicolas; Bartier, Valentin (2019), "Linear Transformations Between Colorings in Chordal Graphs", in Bender, Michael A.; Svensson, Ola; Herman, Grzegorz (eds.), 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, LIPIcs, 144, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 24:1–24:15, doi:10.4230/LIPIcs.ESA.2019.24, S2CID 195791634
- Chung, Fan; Graham, Ron (1998), Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, pp. 97–99.
- Chudnovsky, Maria; Seymour, Paul (2014), "Extending the Gyárfás-Sumner conjecture", Journal of Combinatorial Theory, Series B, 105: 11–16, doi:10.1016/j.jctb.2013.11.002, MR 3171779
- 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, Springer-Verlag, Problem G10.
- Hägglund, Jonas; Steffen, Eckhard (2014), "Petersen-colorings and some families of snarks", Ars Mathematica Contemporanea, 7 (1): 161–173, doi:10.26493/1855-3974.288.11a, MR 3047618, archived from the original on 2016-10-03, retrieved 2016-09-30.
- Jensen, Tommy R.; Toft, Bjarne (1995), "12.20 List-Edge-Chromatic Numbers", Graph Coloring Problems, New York: Wiley-Interscience, pp. 201–202, ISBN 978-0-471-02865-9.
- Molloy, Michael; Reed, Bruce (1998), "A bound on the total chromatic number", Combinatorica, 18 (2): 241–280, CiteSeerX 10.1.1.24.6514, doi:10.1007/PL00009820, MR 1656544, S2CID 9600550.
- Barát, János; Tóth, Géza (2010), "Towards the Albertson Conjecture", Electronic Journal of Combinatorics, 17 (1): R73, arXiv:0909.0413, Bibcode:2009arXiv0909.0413B, doi:10.37236/345.
- Wood, David (January 19, 2009), "Book Thickness of Subdivisions", Open Problem Garden, archived from the original on September 16, 2013, retrieved 2013-02-05.
- Fulek, R.; Pach, J. (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry, 44 (6–7): 345–355, arXiv:1002.3904, doi:10.1007/978-3-642-18469-7_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 978-0-486-31552-2, MR 2047103.
- Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, CiteSeerX 10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645, archived (PDF) from the original on 2016-03-04, retrieved 2016-10-04.
- Nöllenburg, Martin; Prutkin, Roman; Rutter, Ignaz (2016), "On self-approaching and increasing-chord drawings of 3-connected planar graphs", Journal of Computational Geometry, 7 (1): 47–69, arXiv:1409.0315, 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, archived from the original on 2012-08-14, retrieved 2013-03-19.
- 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, North-Holland Mathematics Studies, 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1, ISBN 9780444878038.
- Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7, doi:10.37236/3252.
- 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 Archived 2007-06-13 at the Wayback Machine, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447–1540.
- Lenz, Hanfried; Ringel, Gerhard (1991), "A brief review on Egmont Köhler's mathematical work", Discrete Mathematics, 97 (1–3): 3–16, doi:10.1016/0012-365X(91)90416-Y, MR 1140782
- S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015.
- S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67.
- S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296.
- С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53
- Marc Elliot Glen (2016). "Colourability and word-representability of near-triangulations". arXiv:1605.01688 [math.CO].
- S. Kitaev. On graphs with representation number 3, J. Autom., Lang. and Combin. 18 (2013), 97−112.
- Glen, Marc; Kitaev, Sergey; Pyatkin, Artem (2018). "On the representation number of a crown graph". Discrete Applied Mathematics. 244: 89–93. arXiv:1609.00674. doi:10.1016/j.dam.2018.03.013. S2CID 46925617.
- Conway, John H., Five $1,000 Problems (Update 2017) (PDF), Online Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12
- 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, S2CID 985458, Zbl 1280.05086, archived (PDF) from the original on 2016-03-04, retrieved 2016-09-22.
- Gupta, Anupam; Newman, Ilan; Rabinovich, Yuri; Sinclair, Alistair (2004), "Cuts, trees and -embeddings of graphs", Combinatorica, 24 (2): 233–269, CiteSeerX 10.1.1.698.8978, doi:10.1007/s00493-004-0015-x, MR 2071334, S2CID 46133408
- Pleanmani, Nopparat (2019), "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph", Discrete Mathematics, Algorithms and Applications, 11 (6): 1950068, 7, doi:10.1142/s179383091950068x, MR 4044549
- Spinrad, Jeremy P. (2003), "2. Implicit graph representation", Efficient Graph Representations, pp. 17–30, ISBN 978-0-8218-2815-1.
- "Jorgensen's Conjecture", Open Problem Garden, archived from the original on 2016-11-14, retrieved 2016-11-13.
- Baird, William; Bonato, Anthony (2012), "Meyniel's conjecture on the cop number: a survey", Journal of Combinatorics, 3 (2): 225–238, arXiv:1308.3385, doi:10.4310/JOC.2012.v3.n2.a6, MR 2980752, S2CID 18942362
- Schwenk, Allen (2012), "Some History on the Reconstruction Conjecture" (PDF), Joint Mathematics Meetings, archived (PDF) from the original on 2015-04-09, retrieved 2018-11-26
- Ramachandran, S. (1981), "On a new digraph reconstruction conjecture", Journal of Combinatorial Theory, Series B, 31 (2): 143–149, doi:10.1016/S0095-8956(81)80019-6, MR 0630977
- Seymour's 2nd Neighborhood Conjecture Archived 2019-01-11 at the Wayback Machine, Open Problems in Graph Theory and Combinatorics, Douglas B. West.
- Blokhuis, A.; Brouwer, A. E. (1988), "Geodetic graphs of diameter two", Geometriae Dedicata, 25 (1–3): 527–533, doi:10.1007/BF00191941, MR 0925851, S2CID 189890651
- 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, S2CID 119169562, Zbl 1218.05034.
- 4-flow conjecture Archived 2018-11-26 at the Wayback Machine and 5-flow conjecture Archived 2018-11-26 at the Wayback Machine, Open Problem Garden
- 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, CiteSeerX 10.1.1.159.7029, doi:10.1002/jgt.20565, MR 2864622.
- Ducey, Joshua E. (2017), "On the critical group of the missing Moore graph", Discrete Mathematics, 340 (5): 1104–1109, arXiv:1509.00327, doi:10.1016/j.disc.2016.10.001, MR 3612450, S2CID 28297244
- Fomin, Fedor V.; Høie, Kjartan (2006), "Pathwidth of cubic graphs and exact algorithms", Information Processing Letters, 97 (5): 191–196, doi:10.1016/j.ipl.2005.10.012, MR 2195217
- Khukhro, Evgeny I.; Mazurov, Victor D. (2019), Unsolved Problems in Group Theory. The Kourovka Notebook, arXiv:1401.0300v16
- Aschbacher, Michael (1990), "On Conjectures of Guralnick and Thompson", Journal of Algebra, 135 (2): 277–343, doi:10.1016/0021-8693(90)90292-V
- Hrushovski, Ehud (1989). "Kueker's conjecture for stable theories". Journal of Symbolic Logic. 54 (1): 207–220. doi:10.2307/2275025. JSTOR 2275025.
- Shelah S, Classification Theory, North-Holland, 1990
- Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN 978-1-904987-71-0.
- Peretz, Assaf (2006). "Geometry of forking in simple theories". Journal of Symbolic Logic. 71 (1): 347–359. arXiv:math/0412356. doi:10.2178/jsl/1140641179. S2CID 9380215.
- 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. S2CID 10425739.
- Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
- Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae. 159 (1): 1–50. arXiv:math/9802134. Bibcode:1998math......2134S. doi:10.4064/fm-159-1-1-50. S2CID 8846429.
- Baldwin, John T. (July 24, 2009). Categoricity (PDF). American Mathematical Society. ISBN 978-0-8218-4893-7. Archived (PDF) from the original on July 29, 2010. Retrieved February 20, 2014.
- Shelah, Saharon (2009). "Introduction to classification theory for abstract elementary classes". arXiv:0903.3428. Bibcode:2009arXiv0903.3428S. Cite journal requires
- Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
- Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
- Keisler, HJ (1967). "Ultraproducts which are not saturated". J. Symb. Log. 32 (1): 23–46. doi:10.2307/2271240. JSTOR 2271240.
- Malliaris M, Shelah S, "A dividing line in simple unstable theories." https://arxiv.org/abs/1208.2140 Archived 2017-08-02 at the Wayback Machine
- Conrey, Brian (2016), "Lectures on the Riemann zeta function (book review)", Bulletin of the American Mathematical Society, 53 (3): 507–512, doi:10.1090/bull/1525
- 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.
- Aigner, Martin (2013), Markov's theorem and 100 years of the uniqueness conjecture, Cham: Springer, doi:10.1007/978-3-319-00888-2, ISBN 978-3-319-00887-5, MR 3098784
- Guo, Song; Sun, Zhi-Wei (2005), "On odd covering systems with distinct moduli", Advances in Applied Mathematics, 35 (2): 182–187, arXiv:math/0412217, doi:10.1016/j.aam.2005.01.004, MR 2152886, S2CID 835158
- "Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key". Archived from the original on 2016-03-27. Retrieved 2016-03-18.
- Bruhn, Henning; Schaudt, Oliver (2016). "Newer sums of three cubes". arXiv:1604.07746v1 [math.NT].
- Dobson, J. B. (1 April 2017), "On Lerch's formula for the Fermat quotient", p. 23, arXiv:1103.3907v6 [math.NT]
- Ribenboim, P. (2006). Die Welt der Primzahlen. Springer-Lehrbuch (in German) (2nd ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
- Mazur, Barry (1992), "The topology of rational points", Experimental Mathematics, 1 (1): 35–45, doi:10.1080/10586458.1992.10504244 (inactive 31 May 2021), archived from the original on 2019-04-07, retrieved 2019-04-07CS1 maint: DOI inactive as of May 2021 (link)
- Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric (2006). "The Kadison-Singer problem in mathematics and engineering: A detailed account". In Han, Deguang; Jorgensen, Palle E. T.; Larson, David Royal (eds.). Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida. Contemporary Mathematics. 414. American Mathematical Society. pp. 299–355. doi:10.1090/conm/414/07820. ISBN 978-0-8218-3923-2. Retrieved 24 April 2015.
- Mackenzie, Dana. "Kadison–Singer Problem Solved" (PDF). SIAM News (January/February 2014). Society for Industrial and Applied Mathematics. Archived (PDF) from the original on 23 October 2014. Retrieved 24 April 2015.
- Moreira, Joel; Richter, Florian K.; Robertson, Donald (2019). "A proof of a sumset conjecture of Erdős". Annals of Mathematics. 189 (2): 605–652. arXiv:1803.00498. doi:10.4007/annals.2019.189.2.4. S2CID 119158401.
- Stanley, Richard P. (1994), "A survey of Eulerian posets", in Bisztriczky, T.; McMullen, P.; Schneider, R.; Weiss, A. IviÄ‡ (eds.), Polytopes: abstract, convex and computational (Scarborough, ON, 1993), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 440, Dordrecht: Kluwer Academic Publishers, pp. 301–333, MR 1322068. See in particular p. 316.
- Kalai, Gil (2018-12-25). "Amazing: Karim Adiprasito proved the g-conjecture for spheres!". Archived from the original on 2019-02-16. Retrieved 2019-02-15.
- Santos, Franciscos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics. 176 (1): 383–412. arXiv:1006.2814. doi:10.4007/annals.2012.176.1.7. S2CID 15325169.
- Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica. Extra Volume "Optimization Stories": 75–85. Archived from the original on 2015-04-02. Retrieved 2015-03-25.
- Bowditch, Brian H. (2006). "The angel game in the plane" (PDF). School of Mathematics, University of Southampton: warwick.ac.uk Warwick University. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.
- Kloster, Oddvar. "A Solution to the Angel Problem" (PDF). SINTEF ICT, Postboks 124 Blindern, 0314 Oslo, Norway. Archived from the original (PDF) on 2016-01-07. Retrieved 2016-03-18.CS1 maint: location (link)
- Mathe, Andras (2007). "The Angel of power 2 wins" (PDF). Combinatorics, Probability and Computing. 16 (3): 363–374. Archived (PDF) from the original on 2016-10-13. Retrieved 2016-03-18.
- Gacs, Peter. "THE ANGEL WINS" (PDF). Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.
- Song, Antoine. "Existence of infinitely many minimal hypersurfaces in closed manifolds" (PDF). www.ams.org. Retrieved 19 June 2021.
..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves..
- "Antoine Song | Clay Mathematics Institute".
...Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality
- Wolchover, Natalie (July 11, 2017), "Pentagon Tiling Proof Solves Century-Old Math Problem", Quanta Magazine, archived from the original on August 6, 2017, retrieved July 18, 2017
- Bruhn, Henning; Schaudt, Oliver (2010). "On the Erdos distinct distance problem in the plane". arXiv:1011.4105v3 [math.CO].
- Henle, Frederick V.; Henle, James M. "Squaring the Plane" (PDF). www.maa.org Mathematics Association of America. Archived (PDF) from the original on 2016-03-24. Retrieved 2016-03-18.
- Bruhn, Henning; Schaudt, Oliver (2015). "A formal proof of the Kepler conjecture". arXiv:1501.02155 [math.MG].
- Bruhn, Henning; Schaudt, Oliver (1998). "A proof of the dodecahedral conjecture". arXiv:math/9811079.
- Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 0668845.
- Hartnett, Kevin (19 February 2020). "Rainbow Proof Shows Graphs Have Uniform Parts". Quanta Magazine. Retrieved 2020-02-29.
- Shitov, Yaroslav (May 2019). "Counterexamples to Hedetniemi's conjecture". arXiv:1905.02167 [math.CO].
- Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398. S2CID 117651702.
- "Archived copy" (PDF). Archived from the original (PDF) on 2016-03-03. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
- Bruhn, Henning; Schaudt, Oliver (2005). "Menger's theorem for infinite graphs". arXiv:math/0509397.
- Seigel-Itzkovich, Judy (2008-02-08). "Russian immigrant solves math puzzle". The Jerusalem Post. Retrieved 2015-11-12.
- "Archived copy" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
- Namazi, Hossein; Souto, Juan (2012). "Non-realizability and ending laminations: Proof of the density conjecture". Acta Mathematica. 209 (2): 323–395. doi:10.1007/s11511-012-0088-0.
- 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. 184 (2): 633–682. arXiv:1512.01565. Bibcode:2015arXiv151201565B. doi:10.4007/annals.2016.184.2.7. hdl:1721.1/115568. S2CID 43929329.
- 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].
- Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)", Inventiones Mathematicae, 178 (3): 485–504, Bibcode:2009InMat.178..485K, CiteSeerX 10.1.1.518.4611, doi:10.1007/s00222-009-0205-7, S2CID 14846347
- Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505–586, Bibcode:2009InMat.178..505K, CiteSeerX 10.1.1.228.8022, doi:10.1007/s00222-009-0206-6, S2CID 189820189
- "2011 Cole Prize in Number Theory" (PDF). Notices of the AMS. 58 (4): 610–611. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2015-11-06. Retrieved 2015-11-12.
- Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. Archived (PDF) from the original on 2011-05-10. Retrieved 2016-03-06.
- Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255.
- Lee, Choongbum (2017). "Ramsey numbers of degenerate graphs". Annals of Mathematics. 185 (3): 791–829. arXiv:1505.04773. doi:10.4007/annals.2017.185.3.2. S2CID 7974973.
- Lamb, Evelyn (26 May 2016). "Two-hundred-terabyte maths proof is largest ever". Nature. 534 (7605): 17–18. Bibcode:2016Natur.534...17L. 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 Cube-and-Conquer". In Creignou, N.; Le Berre, D. (eds.). Theory and Applications of Satisfiability Testing – SAT 2016. Lecture Notes in Computer Science. 9710. Springer, [Cham]. pp. 228–245. arXiv:1605.00723. doi:10.1007/978-3-319-40970-2_15. ISBN 978-3-319-40969-6. MR 3534782. S2CID 7912943.
- Linkletter, David (27 December 2019). "The 10 Biggest Math Breakthroughs of 2019". www.popularmechanics.com. Hearst Digital Media. Retrieved 20 June 2021.
- The Conway knot is not slice, Annals of Mathematics, volume 191, issue 2, pp. 581–591
- Graduate Student Solves Decades-Old Conway Knot Problem, Quanta Magazine 19 May 2020
- Bruhn, Henning; Schaudt, Oliver (2012). "The virtual Haken conjecture". arXiv:1204.2810v1 [math.GT].
- Lee, Choongbum (2012). "Embedded minimal tori in S^3 and the Lawson conjecture". arXiv:1203.6597v2 [math.DG].
- Bruhn, Henning; Schaudt, Oliver (2011). "The good pants homology and the Ehrenpreis conjecture". arXiv:1101.1330v4 [math.GT].
- Bruhn, Henning; Schaudt, Oliver (2009). "Rational group ring elements with kernels having irrational dimension". Proceedings of the London Mathematical Society. 107 (6): 1424–1448. arXiv:0909.2360. Bibcode:2009arXiv0909.2360A. doi:10.1112/plms/pdt029. S2CID 115160094.
- Lurie, Jacob (2009). "On the classification of topological field theories". Current Developments in Mathematics. 2008: 129–280. arXiv:0905.0465. Bibcode:2009arXiv0905.0465L. doi:10.4310/cdm.2008.v2008.n1.a3. S2CID 115162503.
- "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF) (Press release). Clay Mathematics Institute. March 18, 2010. Archived from the original on March 22, 2010. Retrieved November 13, 2015.
The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.
- Bruhn, Henning; Schaudt, Oliver (2008). "Completion of the Proof of the Geometrization Conjecture". arXiv:0809.4040 [math.DG].
- Bruhn, Henning; Schaudt, Oliver (2015). "The Erdos discrepancy problem". arXiv:1509.05363v5 [math.CO].
- 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. arXiv:1503.01472. Bibcode:2015arXiv150301472D. doi:10.1186/s40687-015-0044-7. S2CID 43589605.
- Bruhn, Henning; Schaudt, Oliver (2014). "Regularity of Einstein Manifolds and the Codimension 4 Conjecture". arXiv:1406.6534v10 [math.DG].
- "A Long-Sought Proof, Found and Almost Lost". Quanta Magazine. Natalie Wolchover. March 28, 2017. Archived from the original on April 24, 2017. Retrieved May 2, 2017.
- Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. 179 (2): 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6. S2CID 50742102.
- Lee, Choongbum (2011). "A counterexample to Beck's conjecture on the discrepancy of three permutations". arXiv:1104.2922 [cs.DM].
- Voevodsky, Vladimir (1 July 2011). "On motivic cohomology with Z/l-coefficients" (PDF). School of Mathematics Institute for Advanced Study Einstein Drive Princeton, NJ 08540: annals.math.princeton.edu (Princeton University). pp. 401–438. Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.CS1 maint: location (link)
- "Archived copy" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
- "page 359" (PDF). Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.
- "motivic cohomology – Milnor–Bloch–Kato conjecture implies the Beilinson-Lichtenbaum conjecture – MathOverflow". Retrieved 2016-03-18.
- Cilleruelo, Javier (2010). "Generalized Sidon sets". Advances in Mathematics. 225 (5): 2786–2807. doi:10.1016/j.aim.2010.05.010. hdl:10261/31032. S2CID 7385280.
- Bruhn, Henning; Schaudt, Oliver (2009). "A proof of the Kauffman-Harary Conjecture". Algebr. Geom. Topol. 9 (4): 2027–2039. arXiv:0906.1612. Bibcode:2009arXiv0906.1612M. doi:10.2140/agt.2009.9.2027. S2CID 8447495.
- Bruhn, Henning; Schaudt, Oliver (2009). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". arXiv:0910.5501v5 [math.GT].
- Lu, Zhiqin (2007). "Proof of the normal scalar curvature conjecture". arXiv:0711.3510 [math.DG].
- Dencker, Nils (2006), "The resolution of the Nirenberg–Treves conjecture" (PDF), Annals of Mathematics, 163 (2): 405–444, doi:10.4007/annals.2006.163.405, S2CID 16630732, archived (PDF) from the original on 2018-07-20, retrieved 2019-04-07
- "Research Awards", Clay Mathematics Institute, archived from the original on 2019-04-07, retrieved 2019-04-07
- "Archived copy" (PDF). Archived (PDF) from the original on 2016-04-06. Retrieved 2016-03-22.CS1 maint: archived copy as title (link)
- "Fields Medal – Ngô Bảo Châu". International Congress of Mathematicians 2010. ICM. 19 August 2010. Archived from the original on 24 September 2015. Retrieved 2015-11-12.
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 algebro-geometric methods.
- Bruhn, Henning; Schaudt, Oliver (2004). "Tameness of hyperbolic 3-manifolds". arXiv:math/0405568.
- "Graph Theory". Archived from the original on 2016-03-08. Retrieved 2016-03-18.
- Chung, Fan; Greene, Curtis; Hutchinson, Joan (April 2015). "Herbert S. Wilf (1931–2012)". Notices of the AMS. 62 (4): 358. doi:10.1090/noti1247. ISSN 1088-9477. 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. 57 (5): 642–643. May 2010. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.
Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem.
- Bruhn, Henning; Schaudt, Oliver (2004). "The classification of Kleinian surface groups, II: The Ending Lamination Conjecture". arXiv:math/0412006.
- 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/s00454-003-0006-7, MR 1931840, S2CID 40382145
- 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, S2CID 119615076
- "News from 2007". American Mathematical Society. AMS. 31 December 2007. Archived from the original on 17 November 2015. Retrieved 2015-11-13.
The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..."
- Voevodsky, Vladimir (2003). "Reduced power operations in motivic cohomology" (PDF). Publications Mathématiques de l'IHÉS. 98: 1–57. arXiv:math/0107109. CiteSeerX 10.1.1.170.4427. doi:10.1007/s10240-003-0009-z. S2CID 8172797. Archived from the original on 2017-07-28. Retrieved 2016-03-18.
- Savchev, Svetoslav (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297 (1–3): 196–201. doi:10.1016/j.disc.2005.02.018.
- "Archived copy" (PDF). Archived (PDF) from the original on 2016-03-08. Retrieved 2016-03-23.CS1 maint: archived copy as title (link)
- "Archived copy" (PDF). Archived (PDF) from the original on 2016-04-03. Retrieved 2016-03-20.CS1 maint: archived copy as title (link)
- Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2002). "The strong perfect graph theorem". arXiv:math/0212070. doi:10.4007/annals.2006.164.51. Cite journal requires
- Haas, Bertrand. "A Simple Counterexample to Kouchnirenko's Conjecture" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.
- Knight, R. W. (2002), The Vaught Conjecture: A Counterexample, manuscript
- "Archived copy" (PDF). Archived (PDF) from the original on 2016-03-03. Retrieved 2016-03-22.CS1 maint: archived copy as title (link)
- Metsänkylä, Tauno (5 September 2003). "Catalan's conjecture: another old diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. 41 (1): 43–57. doi:10.1090/s0273-0979-03-00993-5. ISSN 0273-0979. Archived (PDF) from the original on 4 March 2016. Retrieved 13 November 2015.
The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu.
- "Archived copy" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
- "Archived copy" (PDF). Archived from the original (PDF) on 2015-09-08. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
- Bruhn, Henning; Schaudt, Oliver (2001). "Deligne's Conjecture on 1-Motives". arXiv:math/0102150.
- Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3-adic exercises", Journal of the American Mathematical Society, 14 (4): 843–939, doi:10.1090/S0894-0347-01-00370-8, ISSN 0894-0347, MR 1839918
- Luca, Florian (2000). "On a conjecture of Erdős and Stewart" (PDF). Mathematics of Computation. 70 (234): 893–897. Bibcode:2001MaCom..70..893L. doi:10.1090/s0025-5718-00-01178-9. Archived (PDF) from the original on 2016-04-02. Retrieved 2016-03-18.
- "Archived copy" (PDF). Archived (PDF) from the original on 2016-04-02. Retrieved 2016-03-20.CS1 maint: archived copy as title (link)
- 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, Bibcode:2003math.....11421C, doi:10.4007/annals.2003.157.545, S2CID 13514070
- Bruhn, Henning; Schaudt, Oliver (1999). "The Honeycomb Conjecture". arXiv:math/9906042.
- Bruhn, Henning; Schaudt, Oliver (1999). "Proof of the gradient conjecture of R. Thom". arXiv:math/9906212.
- Ullmo, E (1998). "Positivité et Discrétion des Points Algébriques des Courbes". Annals of Mathematics. 147 (1): 167–179. arXiv:alg-geom/9606017. doi:10.2307/120987. JSTOR 120987. S2CID 119717506. 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. JSTOR 120986.
- Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 1431-0635, MR 1648105, archived from the original on 2018-04-27, retrieved 2016-03-18
- Norio Iwase (1 November 1998). "Ganea's Conjecture on Lusternik-Schnirelmann 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. 124 (1): 437–449. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424. S2CID 3590991.
- Chen, Zhibo (1996). "Harary's conjectures on integral sum graphs". Discrete Mathematics. 160 (1–3): 241–244. doi:10.1016/0012-365X(95)00163-Q.
Books discussing problems solved since 1995Edit
- Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate. ISBN 978-1-84115-791-7.
- O'Shea, Donal (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9.
- Szpiro, George G. (2003). Kepler's Conjecture. Wiley. ISBN 978-0-471-08601-7.
- Ronan, Mark (2006). Symmetry and the Monster. Oxford. ISBN 978-0-19-280722-9.
Books discussing unsolved problemsEdit
- Chung, Fan; Graham, Ron (1999). Erdös on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 978-1-56881-111-6.
- Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 978-0-387-97506-1.
- Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 978-0-387-20860-2.
- Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 978-0-88385-315-3.
- du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 978-0-06-093558-0.
- Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 978-0-309-08549-6.
- 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 978-0-691-11748-5.
- Ji, Lizhen; Poon, Yat-Sun; Yau, Shing-Tung (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 978-1-57146-278-7.
- Waldschmidt, Michel (2004). "Open Diophantine Problems" (PDF). Moscow Mathematical Journal. 4 (1): 245–305. arXiv:math/0312440. doi:10.17323/1609-4514-2004-4-1-245-305. ISSN 1609-3321. S2CID 11845578. 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 [math.GR].
- The Sverdlovsk Notebook is a collection of unsolved problems in semigroup theory.
- Formulation of unresloved problems for infinite Abelian groups are depicted in the book
- The list of unresolved problems for Combinatorial Geometry are depicted in the book.
- Several dozens of unresolved problems for Combinatorial Geometry are depicted in the book.
- Many unresolved problems for Graph theory are depicted in the article.
- The list of several unresolved problems converning Maler Conjecture are depicted in the book.
- 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
- 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".
- Barry Simon's 15 Problems in Mathematical Physics
- The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1979
- The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1989
- Fuks 1974, p. 47, 88, 116, 134, 158, 159, 186, 210, 242, 243, 292, 318. sfn error: no target: CITEREFFuks1974 (help)
- Boltiansky 1965, p. 83. sfn error: no target: CITEREFBoltiansky1965 (help)
- Grunbaum 1971, p. 6. sfn error: no target: CITEREFGrunbaum1971 (help)
- V. G. Vizing Some unresolved problems for Graph theory // Russian Mathematical Surveys, 23:6(144) (1968), 117–134; Russian Math. Surveys, 23:6 (1968), 125–141
- Sprinjuk 1967, p. 150—154. sfn error: no target: CITEREFSprinjuk1967 (help)