Manymathematical problems have been stated but not yet solved. These problems come from manyareas of mathematics , such astheoretical physics ,computer science ,algebra ,analysis ,combinatorics ,algebraic ,differential ,discrete andEuclidean geometries ,graph theory ,group theory ,model theory ,number theory ,set theory ,Ramsey theory ,dynamical systems , andpartial differential equations . Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as theMillennium Prize Problems , receive considerable attention.
This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance.
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.
TheRiemann zeta function , subject of theRiemann hypothesis [ 12] Millennium Prize Problems [ edit ] Of the original sevenMillennium Prize Problems listed by theClay Mathematics Institute in 2000, six remain unsolved to date:[ 6]
The seventh problem, thePoincaré conjecture , was solved byGrigori Perelman in 2003.[ 13] smooth four-dimensional Poincaré conjecture —that is, whether afour -dimensionaltopological sphere can have two or more inequivalentsmooth structures —is unsolved.[ 14]
In theBloch sphere representation of aqubit , aSIC-POVM forms aregular tetrahedron . Zauner conjectured that analogous structures exist in complexHilbert spaces of all finite dimensions. Birch–Tate conjecture on the relation between the order of thecenter of theSteinberg group of thering of integers of anumber field to the field'sDedekind zeta function .Casas-Alvero conjecture : if a polynomial of degreed {\displaystyle d} field K {\displaystyle K} characteristic 0 {\displaystyle 0} d − 1 {\displaystyle d-1} f {\displaystyle f} d {\displaystyle d} Connes embedding problem inVon Neumann algebra theoryCrouzeix's conjecture : thematrix norm of a complex functionf {\displaystyle f} A {\displaystyle A} supremum of| f ( z ) | {\displaystyle |f(z)|} field of values ofA {\displaystyle A} Determinantal conjecture on thedeterminant of the sum of twonormal matrices .Eilenberg–Ganea conjecture : a group withcohomological dimension 2 also has a 2-dimensionalEilenberg–MacLane space K ( G , 1 ) {\displaystyle K(G,1)} Farrell–Jones conjecture on whether certainassembly maps areisomorphisms .Finite lattice representation problem : is every finitelattice isomorphic to thecongruence lattice of some finitealgebra ?[ 22] Goncharov conjecture on thecohomology of certainmotivic complexes .Green's conjecture : theClifford index of a non-hyperelliptic curve is determined by the extent to which it, as acanonical curve , haslinear syzygies .Grothendieck–Katz p-curvature conjecture : a conjecturedlocal–global principle forlinear ordinary differential equations .Hadamard conjecture : for every positive integerk {\displaystyle k} Hadamard matrix of order4 k {\displaystyle 4k} Williamson conjecture : the problem of finding Williamson matrices, which can be used to construct Hadamard matrices.Hadamard's maximal determinant problem : what is the largestdeterminant of a matrix with entries all equal to 1 or −1?Hilbert's fifteenth problem : putSchubert calculus on a rigorous foundation.Hilbert's sixteenth problem : what are the possible configurations of theconnected components ofM-curves ?Homological conjectures in commutative algebra Jacobson's conjecture : the intersection of all powers of theJacobson radical of a left-and-rightNoetherian ring is precisely 0.Kaplansky's conjectures Köthe conjecture : if a ring has nonil ideal other than{ 0 } {\displaystyle \{0\}} one-sided ideal other than{ 0 } {\displaystyle \{0\}} Monomial conjecture onNoetherian local rings Existence ofperfect cuboids and associatedcuboid conjectures Pierce–Birkhoff conjecture : every piecewise-polynomialf : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } Rota's basis conjecture : for matroids of rankn {\displaystyle n} n {\displaystyle n} B i {\displaystyle B_{i}} n × n {\displaystyle n\times n} B i {\displaystyle B_{i}} Serre's conjecture II : ifG {\displaystyle G} simply connected semisimple algebraic group over a perfectfield ofcohomological dimension at most2 {\displaystyle 2} Galois cohomology setH 1 ( F , G ) {\displaystyle H^{1}(F,G)} Serre's positivity conjecture that ifR {\displaystyle R} regular local ring , andP , Q {\displaystyle P,Q} prime ideals ofR {\displaystyle R} dim ( R / P ) + dim ( R / Q ) = dim ( R ) {\displaystyle \dim(R/P)+\dim(R/Q)=\dim(R)} χ ( R / P , R / Q ) > 0 {\displaystyle \chi (R/P,R/Q)>0} Uniform boundedness conjecture for rational points : doalgebraic curves ofgenus g ≥ 2 {\displaystyle g\geq 2} number fields K {\displaystyle K} N ( K , g ) {\displaystyle N(K,g)} K {\displaystyle K} rational points ?Wild problems : problems involving classification of pairs ofn × n {\displaystyle n\times n} Zariski–Lipman conjecture : for acomplex algebraic variety V {\displaystyle V} coordinate ring R {\displaystyle R} derivations ofR {\displaystyle R} free module overR {\displaystyle R} V {\displaystyle V} smooth .Zauner's conjecture: doSIC-POVMs exist in all dimensions? Zilber–Pink conjecture that ifX {\displaystyle X} Shimura variety orsemiabelian variety defined overC {\displaystyle \mathbb {C} } V ⊆ X {\displaystyle V\subseteq X} V {\displaystyle V} Thefree Burnside group B ( 2 , 3 ) {\displaystyle B(2,3)} Cayley graph , shown here, each of its 27 elements is represented by a vertex. The question of which other groupsB ( m , n ) {\displaystyle B(m,n)} Representation theory [ edit ] A detail of theMandelbrot set . It is not known whether the Mandelbrot set islocally connected or not. Combinatorial games [ edit ] Games with imperfect information [ edit ] Abundance conjecture : if thecanonical bundle of aprojective variety withKawamata log terminal singularities isnef , then it is semiample.Bass conjecture on thefinite generation of certainalgebraic K-groups .Bass–Quillen conjecture relatingvector bundles over aregular Noetherian ring and over thepolynomial ring A [ t 1 , … , t n ] {\displaystyle A[t_{1},\ldots ,t_{n}]} Deligne conjecture : any one of numerous named forPierre Deligne .Dixmier conjecture : anyendomorphism of aWeyl algebra is anautomorphism .Fröberg conjecture on theHilbert functions of a set of forms.Fujita conjecture regarding the line bundleK M ⊗ L ⊗ m {\displaystyle K_{M}\otimes L^{\otimes m}} positive holomorphic line bundle L {\displaystyle L} compact complex manifold M {\displaystyle M} canonical line bundle K M {\displaystyle K_{M}} M {\displaystyle M} General elephant problem : dogeneral elephants have at mostDu Val singularities ?Hartshorne's conjectures[ 43] Inspherical orhyperbolic geometry , must polyhedra with the same volume andDehn invariant bescissors-congruent ?[ 44] Jacobian conjecture : if apolynomial mapping over acharacteristic -0 field has a constant nonzeroJacobian determinant , then it has aregular (i.e. with polynomial components) inverse function.Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence betweenGromov–Witten theory andDonaldson–Thomas theory [ 45] Nagata's conjecture on curves , specifically the minimal degree required for aplane algebraic curve to pass through a collection of very general points with prescribedmultiplicities .Nagata–Biran conjecture that ifX {\displaystyle X} algebraic surface andL {\displaystyle L} ample line bundle onX {\displaystyle X} d {\displaystyle d} r {\displaystyle r} Seshadri constant satisfiesε ( p 1 , … , p r ; X , L ) = d / r {\displaystyle \varepsilon (p_{1},\ldots ,p_{r};X,L)=d/{\sqrt {r}}} Nakai conjecture : if acomplex algebraic variety has a ring ofdifferential operators generated by its containedderivations , then it must besmooth .Parshin's conjecture : the higheralgebraic K-groups of anysmooth projective variety defined over afinite field must vanish up to torsion.Section conjecture on splittings ofgroup homomorphisms fromfundamental groups of completesmooth curves over finitely-generatedfields k {\displaystyle k} Galois group ofk {\displaystyle k} Standard conjectures on algebraic cyclesTate conjecture on the connection betweenalgebraic cycles onalgebraic varieties andGalois representations onétale cohomology groups .Virasoro conjecture : a certaingenerating function encoding theGromov–Witten invariants of asmooth projective variety is fixed by an action of half of theVirasoro algebra .Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity ofvarieties atsingular points [ 46] Are infinite sequences offlips possible in dimensions greater than 3? Resolution of singularities in characteristicp {\displaystyle p} Covering and packing [ edit ] Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover abounded n -dimensional set.Thecovering 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?[ 47] TheErdős–Oler conjecture : whenn {\displaystyle n} triangular number , packingn − 1 {\displaystyle n-1} n {\displaystyle n} [ 48] Thedisk covering problem about finding the smallestreal number r ( n ) {\displaystyle r(n)} n {\displaystyle n} disks of radiusr ( n ) {\displaystyle r(n)} unit disk . Thekissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[ 49] Reinhardt's conjecture : the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[ 50] 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?[ 51] Ulam's packing conjecture about the identity of the worst-packing convex solid[ 52] TheTammes problem for numbers of nodes greater than 14 (except 24).[ 53] Differential geometry [ edit ] In three dimensions, thekissing 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 aregular icosahedron .) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24. Are the two Meissner tetrahedra the minimum-volume three-dimensional shapes of constant width?[ 83] Non-Euclidean geometry [ edit ] Algebraic graph theory [ edit ] 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. Graph drawing and embedding [ edit ] Restriction of graph parameters [ edit ] Does there exist aconference graph for every number of verticesv > 1 {\displaystyle v>1} v ≡ 1 mod 4 {\displaystyle v\equiv 1{\bmod {4}}} v {\displaystyle v} [ 117] Conway's 99-graph problem : does there exist astrongly regular graph with parameters( 99 , 14 , 1 , 2 ) {\displaystyle (99,14,1,2)} [ 118] Degree diameter problem : given two positive integersd , k {\displaystyle d,k} k {\displaystyle k} d {\displaystyle d} Jørgensen's conjecture that every 6-vertex-connectedK 6 {\displaystyle K_{6}} apex graph [ 119] Does aMoore graph with girth 5 and degree 57 exist?[ 120] Do there exist infinitely manystrongly regular geodetic graphs , or any strongly regular geodetic graphs that are not Moore graphs?[ 121] Word-representation of graphs [ edit ] Miscellaneous graph theory [ edit ] Delta-conjecture (1978): consider acomplete graph ( V ; E 1 , … , E d ) {\displaystyle (V;E_{1},\dots ,E_{d})} d {\displaystyle d} Δ {\displaystyle \Delta } ( V , E i ) , i = 1 , … , d {\displaystyle (V,E_{i}),i=1,\dots ,d} independent vertex-set such that the intersection of the obtainedd {\displaystyle d} [ 146] [ 147] Theimbalance conjecture : If the imbalance for each edge of a graph is at least 1, is the multiset of all edge imbalances alwaysgraphic ?[ 148] Theimplicit graph conjecture on the existence of implicit representations for slowly-growinghereditary families of graphs [ 149] Ryser's conjecture relating the maximummatching size and minimumtransversal size inhypergraphs Thesecond 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?[ 150] Sidorenko's conjecture onhomomorphism densities of graphs ingraphons Teschner'sbondage number conjecture: is the bondage number of a graph always less than or equal to 3/2 times its maximumdegree ?[ 151] Tutte's conjectures: Woodall's conjecture that the minimum number of edges in adicut of adirected graph is equal to the maximum number of disjointdijoins .Model theory and formal languages [ edit ] TheCherlin–Zilber conjecture : A simple group whose first-order theory isstable inℵ 0 {\displaystyle \aleph _{0}} Generalized star height problem : can allregular languages be expressed usinggeneralized regular expressions with limited nesting depths ofKleene stars ?For which number fields doesHilbert's tenth problem hold? Kueker's conjecture[ 154] The main gap conjecture, e.g. for uncountablefirst order theories , forAECs , and forℵ 1 {\displaystyle \aleph _{1}} [ 155] Shelah's categoricity conjecture forL ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} [ 155] Shelah's eventual categoricity conjecture: For every cardinalλ {\displaystyle \lambda } μ ( λ ) {\displaystyle \mu (\lambda )} AEC K with LS(K )≤ λ {\displaystyle {}\leq \lambda } μ ( λ ) {\displaystyle \mu (\lambda )} μ ( λ ) {\displaystyle \mu (\lambda )} [ 155] [ 156] The stable field conjecture: every infinite field with astable first-order theory is separably closed. The stable forking conjecture for simple theories[ 157] Tarski's exponential function problem : is thetheory of thereal numbers with theexponential function decidable ?The universality problem forC -free graphs: For which finite setsC of graphs does the class ofC -free countable graphs have a universal member under strong embeddings?[ 158] The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[ 159] Vaught conjecture : the number ofcountable models of afirst-order complete theory in a countablelanguage is either finite,ℵ 0 {\displaystyle \aleph _{0}} 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} AssumeK is the class of models of a countable first order theory omitting countably manytypes . IfK has a model of cardinalityℵ ω 1 {\displaystyle \aleph _{\omega _{1}}} [ 160] Do theHenson graphs have thefinite model property ? Does a finitely presented homogeneous structure for a finite relational language have finitely manyreducts ? Does there exist ano-minimal first order theory with a trans-exponential (rapid growth) function? If the class of atomic models of a complete first order theory iscategorical in theℵ n {\displaystyle \aleph _{n}} [ 161] [ 162] Is every infinite, minimal field of characteristic zeroalgebraically closed ? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.) Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?[ 163] Is the theory of the field of Laurent series overZ p {\displaystyle \mathbb {Z} _{p}} decidable ? of the field of polynomials overC {\displaystyle \mathbb {C} } Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[ 164] Determine the structure of Keisler's order.[ 165] [ 166] What is the nature of theproof-theoretic ordinal (the smallest ordinal a theory cannot prove well-founded) forsecond-order arithmetic ,ZFC , or stronger theories?[ 167] 6 is aperfect 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 is odd. Büchi's problem on sufficiently large sequences of square numbers with constant second difference.Carmichael's totient function conjecture : do all values ofEuler's totient function havemultiplicity greater than1 {\displaystyle 1} Catalan–Dickson conjecture on aliquot sequences : noaliquot sequences are infinite but non-repeating.Exponent pair conjecture : for allε > 0 {\displaystyle \varepsilon >0} ( ε , 1 / 2 + ε ) {\displaystyle (\varepsilon ,1/2+\varepsilon )} exponent pair ?TheGauss circle problem : how far can the number of integer points in a circle centered at the origin be from the area of the circle? Grimm's conjecture : each element of a set of consecutivecomposite numbers can be assigned a distinctprime number that divides it.Hall's conjecture : for anyε > 0 {\displaystyle \varepsilon >0} c ( ε ) {\displaystyle c(\varepsilon )} y 2 = x 3 {\displaystyle y^{2}=x^{3}} | y 2 − x 3 | > c ( ε ) x 1 / 2 − ε {\displaystyle |y^{2}-x^{3}|>c(\varepsilon )x^{1/2-\varepsilon }} Lehmer's totient problem : ifϕ ( n ) {\displaystyle \phi (n)} n − 1 {\displaystyle n-1} n {\displaystyle n} Magic square of squares : is there a 3x3 magic square composed of distinct perfect squares?Mahler's 3/2 problem that no real numberx {\displaystyle x} x ( 3 / 2 ) n {\displaystyle x(3/2)^{n}} 1 / 2 {\displaystyle 1/2} n {\displaystyle n} Newman's conjecture : thepartition function satisfies any arbitrary congruence infinitely often.Scholz conjecture : the length of the shortestaddition chain producing2 n − 1 {\displaystyle 2^{n}-1} n − 1 {\displaystyle n-1} n {\displaystyle n} Singmaster's conjecture : is there a finite upper bound on the multiplicities of the entries greater than 1 inPascal's triangle ?[ 168] Are there infinitely manyperfect numbers ? Do anyodd perfect numbers exist? Doquasiperfect numbers exist? Do any non-power of 2almost perfect numbers exist? Are there 65, 66, or 67idoneal numbers ? Are there any pairs ofamicable numbers which have opposite parity? Are there any pairs ofbetrothed numbers which have same parity? Are there any pairs ofrelatively prime amicable numbers ? Are there infinitely many pairs ofamicable numbers ? Are there infinitely manybetrothed numbers ? Are there infinitely manyGiuga numbers ? Do anyLychrel numbers exist in base 10? Do any oddnoncototients exist? Do any oddweird numbers exist? Do any(2, 5)-perfect numbers exist? Do anyTaxicab(5, 2, n) exist forn > 1? Is there acovering system with odd distinct moduli?[ 169] Isπ {\displaystyle \pi } normal number (i.e., is each digit 0–9 equally frequent)?[ 170] Are allirrational algebraic numbers normal? Is 10 asolitary number ? Additive number theory [ edit ] Algebraic number theory [ edit ] Analytic number theory [ edit ] Arithmetic geometry [ edit ] Computational number theory [ edit ] Diophantine approximation and transcendental number theory [ edit ] The area of the blue region converges to theEuler–Mascheroni constant , which may or may not be a rational number. Littlewood conjecture : for any two real numbersα , β {\displaystyle \alpha ,\beta } lim inf n → ∞ n ‖ n α ‖ ‖ n β ‖ = 0 {\displaystyle \liminf _{n\rightarrow \infty }n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0} ‖ x ‖ {\displaystyle \Vert x\Vert } x {\displaystyle x} Schanuel's conjecture on thetranscendence degree of certainfield extensions of the rational numbers.[ 173] π {\displaystyle \pi } e {\displaystyle e} algebraically independent ? Which nontrivial combinations oftranscendental numbers (such ase + π , e π , π e , π π , e e {\displaystyle e+\pi ,e\pi ,\pi ^{e},\pi ^{\pi },e^{e}} [ 174] [ 175] Thefour exponentials conjecture : the transcendence of at least one of four exponentials of combinations of irrationals[ 173] AreEuler's constant γ {\displaystyle \gamma } Catalan's constant G {\displaystyle G} Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} [ 176] [ 177] Which transcendental numbers are(exponential) periods ?[ 178] How well cannon-quadratic irrational numbers be approximated? What is theirrationality measure of specific (suspected) transcendental numbers such asπ {\displaystyle \pi } γ {\displaystyle \gamma } [ 177] Which irrational numbers havesimple continued fraction terms whosegeometric mean converges toKhinchin's constant ?[ 179] Hartmanis–Stearns conjecture Diophantine equations [ edit ] Beal's conjecture : for all integral solutions toA x + B y = C z {\displaystyle A^{x}+B^{y}=C^{z}} x , y , z > 2 {\displaystyle x,y,z>2} A , B , C {\displaystyle A,B,C} Brocard's problem : are there any integer solutions ton ! + 1 = m 2 {\displaystyle n!+1=m^{2}} n = 4 , 5 , 7 {\displaystyle n=4,5,7} Congruent number problem (a corollary toBirch and Swinnerton-Dyer conjecture , perTunnell's theorem ): determine precisely what rational numbers arecongruent numbers .Erdős–Moser problem: is1 1 + 2 1 = 3 1 {\displaystyle 1^{1}+2^{1}=3^{1}} Erdős–Moser equation ? Erdős–Straus conjecture : for everyn ≥ 2 {\displaystyle n\geq 2} x , y , z {\displaystyle x,y,z} 4 / n = 1 / x + 1 / y + 1 / z {\displaystyle 4/n=1/x+1/y+1/z} Fermat–Catalan conjecture : there are finitely many distinct solutions( a m , b n , c k ) {\displaystyle (a^{m},b^{n},c^{k})} a m + b n = c k {\displaystyle a^{m}+b^{n}=c^{k}} a , b , c {\displaystyle a,b,c} coprime integers andm , n , k {\displaystyle m,n,k} 1 / m + 1 / n + 1 / k < 1 {\displaystyle 1/m+1/n+1/k<1} Goormaghtigh conjecture on solutions to( x m − 1 ) / ( x − 1 ) = ( y n − 1 ) / ( y − 1 ) {\displaystyle (x^{m}-1)/(x-1)=(y^{n}-1)/(y-1)} x > y > 1 {\displaystyle x>y>1} m , n > 2 {\displaystyle m,n>2} Theuniqueness conjecture for Markov numbers [ 180] Markov number is the largest number in exactly one normalized solution to the MarkovDiophantine equation . Pillai's conjecture : for anyA , B , C {\displaystyle A,B,C} A x m − B y n = C {\displaystyle Ax^{m}-By^{n}=C} m , n {\displaystyle m,n} 2 {\displaystyle 2} Which integers can be written as thesum of three perfect cubes ?[ 181] Can every integer be written as a sum of four perfect cubes?
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.Agoh–Giuga conjecture on theBernoulli numbers thatp {\displaystyle p} p B p − 1 ≡ − 1 ( mod p ) {\displaystyle pB_{p-1}\equiv -1{\pmod {p}}} Agrawal's conjecture that givencoprime positive integers n {\displaystyle n} r {\displaystyle r} ( X − 1 ) n ≡ X n − 1 ( mod n , X r − 1 ) {\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,X^{r}-1}}} n {\displaystyle n} n 2 ≡ 1 ( mod r ) {\displaystyle n^{2}\equiv 1{\pmod {r}}} Artin's conjecture on primitive roots that if an integer is neither a perfect square nor− 1 {\displaystyle -1} primitive root modulo infinitely manyprime numbers p {\displaystyle p} Brocard's conjecture : there are always at least4 {\displaystyle 4} prime numbers between consecutive squares of prime numbers, aside from2 2 {\displaystyle 2^{2}} 3 2 {\displaystyle 3^{2}} Bunyakovsky conjecture : if an integer-coefficient polynomialf {\displaystyle f} f ( x ) {\displaystyle f(x)} x {\displaystyle x} f ( x ) {\displaystyle f(x)} Catalan's Mersenne conjecture : someCatalan–Mersenne number is composite and thus all Catalan–Mersenne numbers are composite after some point.Dickson's conjecture : for a finite set of linear formsa 1 + b 1 n , … , a k + b k n {\displaystyle a_{1}+b_{1}n,\ldots ,a_{k}+b_{k}n} b i ≥ 1 {\displaystyle b_{i}\geq 1} n {\displaystyle n} prime , unless there is somecongruence condition preventing it.Dubner's conjecture: every even number greater than4208 {\displaystyle 4208} primes which both have atwin . Elliott–Halberstam conjecture on the distribution ofprime numbers inarithmetic progressions .Erdős–Mollin–Walsh conjecture : no three consecutive numbers are allpowerful .Feit–Thompson conjecture : for all distinctprime numbers p {\displaystyle p} q {\displaystyle q} ( p q − 1 ) / ( p − 1 ) {\displaystyle (p^{q}-1)/(p-1)} ( q p − 1 ) / ( q − 1 ) {\displaystyle (q^{p}-1)/(q-1)} Fortune's conjecture that noFortunate number is composite. TheGaussian moat problem: is it possible to find an infinite sequence of distinctGaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded? Gillies' conjecture on the distribution ofprime divisors ofMersenne numbers .Landau's problems Problems associated toLinnik's theorem New Mersenne conjecture : for any oddnatural number p {\displaystyle p} p = 2 k ± 1 {\displaystyle p=2^{k}\pm 1} p = 4 k ± 3 {\displaystyle p=4^{k}\pm 3} 2 p − 1 {\displaystyle 2^{p}-1} ( 2 p + 1 ) / 3 {\displaystyle (2^{p}+1)/3} Polignac's conjecture : for all positive even numbersn {\displaystyle n} prime gaps of sizen {\displaystyle n} Schinzel's hypothesis H that for every finite collection{ f 1 , … , f k } {\displaystyle \{f_{1},\ldots ,f_{k}\}} irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integersn {\displaystyle n} f 1 ( n ) , … , f k ( n ) {\displaystyle f_{1}(n),\ldots ,f_{k}(n)} primes , or there is some fixed divisorm > 1 {\displaystyle m>1} n {\displaystyle n} f i ( n ) {\displaystyle f_{i}(n)} Selfridge's conjecture : is 78,557 the lowestSierpiński number ?Does theconverse of Wolstenholme's theorem hold for all natural numbers? Are allEuclid numbers square-free ? Are allFermat numbers square-free ? Are allMersenne numbers of prime indexsquare-free ? Are there any compositec satisfying 2c − 1c 2 )? Are there anyWall–Sun–Sun primes ? Are there anyWieferich primes in base 47? Are there infinitely manybalanced primes ? Are there infinitely manycluster primes ? Are there infinitely manycousin primes ? Are there infinitely manyCullen primes ? Are there infinitely manyEuclid primes ? Are there infinitely manyFibonacci primes ? Are there infinitely manyKummer primes ? Are there infinitely many Kynea primes? Are there infinitely manyLucas primes ? Are there infinitely manyMersenne primes (Lenstra–Pomerance–Wagstaff conjecture ); equivalently, infinitely many evenperfect numbers ? Are there infinitely manyNewman–Shanks–Williams primes ? Are there infinitely manypalindromic primes to every base? Are there infinitely manyPell primes ? Are there infinitely manyPierpont primes ? Are there infinitely manyprime quadruplets ? Are there infinitely manyprime triplets ? Siegel's conjecture : are there infinitely many regular primes, and if so is theirnatural density as a subset of all primese − 1 / 2 {\displaystyle e^{-1/2}} Are there infinitely manysexy primes ? Are there infinitely manysafe and Sophie Germain primes ? Are there infinitely manyWagstaff primes ? Are there infinitely manyWieferich primes ? Are there infinitely manyWilson primes ? Are there infinitely manyWolstenholme primes ? Are there infinitely manyWoodall primes ? Can a primep satisfy2 p − 1 ≡ 1 ( mod p 2 ) {\displaystyle 2^{p-1}\equiv 1{\pmod {p^{2}}}} 3 p − 1 ≡ 1 ( mod p 2 ) {\displaystyle 3^{p-1}\equiv 1{\pmod {p^{2}}}} [ 182] Does every prime number appear in theEuclid–Mullin sequence ? What is the smallestSkewes's number ? For any given integera > 0, are there infinitely manyLucas–Wieferich primes associated with the pair (a , −1)? (Specially, whena = 1, this is the Fibonacci-Wieferich primes, and whena = 2, this is the Pell-Wieferich primes) For any given integera > 0, are there infinitely many primesp such thata p − 1p 2 )?[ 183] For any given integerb which is not a perfect power and not of the form −4k 4 for integerk , are there infinitely manyrepunit primes to baseb ? For any given integersk ≥ 1 , b ≥ 2 , c ≠ 0 {\displaystyle k\geq 1,b\geq 2,c\neq 0} gcd(k ,c ) = 1 andgcd(b ,c ) = 1, are there infinitely many primes of the form( k × b n + c ) / gcd ( k + c , b − 1 ) {\displaystyle (k\times b^{n}+c)/\gcd(k+c,b-1)} n ≥ 1? Is everyFermat number 2 2 n + 1 {\displaystyle 2^{2^{n}}+1} n > 4 {\displaystyle n>4} Is 509,203 the lowestRiesel number ? Note: The following conjectures are expressed in thefirst-order language ofaxiomatic set theory and, unless stated otherwise, are here taken to be overZermelo-Frankel set theory , possibly withChoice . In particular, the conjecture'sindependence may not be open in set theories with a wider or conflicting class ofmodels , such as the variousconstructive resp.non-wellfounded set theories, etc.
Theunknotting problem asks whether there is an efficient algorithm to identify when the shape presented in aknot diagram is actually theunknot . Problems solved since 1995 [ edit ] Ricci flow , here illustrated with a 2D manifold, was the key tool inGrigori Perelman 'ssolution of the Poincaré conjecture .Mizohata–Takeuchi conjecture (Hannah Cairo , 2025).[ 192] [ 193] Kadison–Singer problem (Adam Marcus ,Daniel Spielman andNikhil Srivastava , 2013)[ 194] [ 195] Feichtinger's conjecture , Anderson's paving conjectures, Weaver's discrepancy theoreticK S r {\displaystyle KS_{r}} K S r ′ {\displaystyle KS'_{r}} R ε {\displaystyle R_{\varepsilon }} Ahlfors measure conjecture (Ian Agol , 2004)[ 196] Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)[ 197] Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)[ 198] 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)[ 199] [ 200] Hirsch conjecture (Francisco Santos Leal , 2010)[ 201] [ 202] Gessel's lattice path conjecture (Manuel Kauers ,Christoph Koutschan , andDoron Zeilberger , 2009)[ 203] Stanley–Wilf conjecture (Gábor Tardos andAdam Marcus , 2004)[ 204] Kemnitz's conjecture (Christian Reiher , 2003, Carlos di Fiore, 2003)[ 205] Cameron–Erdős conjecture (Ben J. Green , 2003, Alexander Sapozhenko, 2003)[ 206] [ 207] Carathéodory conjecture for surfaces of smoothnessC 4 {\displaystyle C^{4}} [ 217] Einstein problem (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2024)[ 218] Maximal rank conjecture (Eric Larson, 2018)[ 219] Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018)[ 220] Yau's conjecture (Antoine Song , 2018)[ 221] [ 222] Pentagonal tiling (Michaël Rao, 2017)[ 223] Willmore conjecture (Fernando Codá Marques andAndré Neves , 2012)[ 224] Erdős distinct distances problem (Larry Guth ,Nets Hawk Katz , 2011)[ 225] Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)[ 226] Tameness conjecture (Ian Agol , 2004)[ 196] Ending lamination theorem (Jeffrey F. Brock ,Richard D. Canary ,Yair N. Minsky , 2004)[ 227] Carpenter's rule problem (Robert Connelly ,Erik Demaine , Günter Rote, 2003)[ 228] Lambda g conjecture (Carel Faber andRahul Pandharipande , 2003)[ 229] Nagata's conjecture (Ivan Shestakov, Ualbai Umirbaev, 2003)[ 230] Double bubble conjecture (Michael Hutchings ,Frank Morgan , Manuel Ritoré, Antonio Ros, 2002)[ 231] Kahn–Kalai conjecture (Jinyoung Park and Huy Tuan Pham, 2022)[ 238] Blankenship–Oporowski conjecture on the book thickness of subdivisions (Vida Dujmović ,David Eppstein , Robert Hickingbotham,Pat Morin , andDavid Wood , 2021)[ 239] Ringel's conjecture that the complete graphK 2 n + 1 {\displaystyle K_{2n+1}} 2 n + 1 {\displaystyle 2n+1} n {\displaystyle n} Benny Sudakov , Alexey Pokrovskiy, 2020)[ 240] [ 241] Disproof ofHedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)[ 242] Kelmans–Seymour conjecture (Dawei He, Yan Wang, and Xingxing Yu, 2020)[ 243] [ 244] [ 245] [ 246] Goldberg–Seymour conjecture (Guantao Chen, Guangming Jing, and Wenan Zang, 2019)[ 247] Babai's problem (Alireza Abdollahi, Maysam Zallaghi, 2015)[ 248] Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)Alon–Saks–Seymour conjecture (Hao Huang,Benny Sudakov , 2012)Read–Hoggar conjecture (June Huh , 2009)[ 249] Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)[ 250] Erdős–Menger conjecture (Ron Aharoni , Eli Berger 2007)[ 251] Road coloring conjecture (Avraham Trahtman , 2007)[ 252] Robertson–Seymour theorem (Neil Robertson ,Paul Seymour , 2004)[ 253] Strong perfect graph conjecture (Maria Chudnovsky ,Neil Robertson ,Paul Seymour andRobin Thomas , 2002)[ 254] Toida's conjecture (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001)[ 255] Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996)[ 256] André–Oort conjecture (Jonathan Pila , Ananth Shankar,Jacob Tsimerman , 2021)[ 260] Duffin–Schaeffer theorem (Dimitris Koukoulopoulos ,James Maynard , 2019)Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain , Ciprian Demeter,Larry Guth , 2015)[ 261] Goldbach's weak conjecture (Harald Helfgott , 2013)[ 262] [ 263] [ 264] Existence of bounded gaps between arbitrarily large primes (Yitang Zhang ,Polymath8 ,James Maynard , 2013)[ 265] [ 266] [ 267] Sidon set problem (Javier Cilleruelo,Imre Z. Ruzsa , and Carlos Vinuesa, 2010)[ 268] Serre's modularity conjecture (Chandrashekhar Khare andJean-Pierre Wintenberger , 2008)[ 269] [ 270] [ 271] Green–Tao theorem (Ben J. Green andTerence Tao , 2004)[ 272] Catalan's conjecture (Preda Mihăilescu , 2002)[ 273] Erdős–Graham problem (Ernest S. Croot III , 2000)[ 274] Theoretical computer science [ edit ] Deciding whether theConway knot is aslice knot (Lisa Piccirillo , 2020)[ 282] [ 283] Virtual Haken conjecture (Ian Agol , Daniel Groves, Jason Manning, 2012)[ 284] Daniel Wise alsovirtually fibered conjecture )Hsiang–Lawson's conjecture (Simon Brendle , 2012)[ 285] Ehrenpreis conjecture (Jeremy Kahn ,Vladimir Markovic , 2011)[ 286] Atiyah conjecture for groups with finite subgroups of unbounded order (Austin, 2009)[ 287] Cobordism hypothesis (Jacob Lurie , 2008)[ 288] Spherical space form conjecture (Grigori Perelman , 2006)Poincaré conjecture (Grigori Perelman , 2002)[ 289] Geometrization conjecture (Grigori Perelman ,[ 289] [ 290] Nikiel's conjecture (Mary Ellen Rudin , 1999)[ 291] Disproof of theGanea conjecture (Iwase, 1997)[ 292] Erdős discrepancy problem (Terence Tao , 2015)[ 293] Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin,Ken Ono , 2015)[ 294] Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (Jeff Cheeger , Aaron Naber, 2014)[ 295] Gaussian correlation inequality (Thomas Royen , 2014)[ 296] Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman,Aleksandar Nikolov , 2011)[ 297] Bloch–Kato conjecture (Vladimir Voevodsky , 2011)[ 298] Quillen–Lichtenbaum conjecture and by work ofThomas Geisser andMarc Levine (2001) alsoBeilinson–Lichtenbaum conjecture [ 299] [ 300] : 359 [ 301] Kauffman–Harary conjecture (Thomas Mattman, Pablo Solis, 2009)[ 302] Surface subgroup conjecture (Jeremy Kahn ,Vladimir Markovic , 2009)[ 303] Normal scalar curvature conjecture and theBöttcher–Wenzel conjecture (Zhiqin Lu, 2007)[ 304] Nirenberg–Treves conjecture (Nils Dencker , 2005)[ 305] [ 306] Lax conjecture (Adrian Lewis ,Pablo Parrilo , Motakuri Ramana, 2005)[ 307] TheLanglands–Shelstad fundamental lemma (Ngô Bảo Châu andGérard Laumon , 2004)[ 308] Milnor conjecture (Vladimir Voevodsky , 2003)[ 309] Kirillov's conjecture (Ehud Baruch, 2003)[ 310] Kouchnirenko's conjecture (Bertrand Haas, 2002)[ 311] n ! conjectureMark Haiman , 2001)[ 312] Macdonald positivity conjecture )Kato's conjecture (Pascal Auscher ,Steve Hofmann ,Michael Lacey ,Alan McIntosh , and Philipp Tchamitchian, 2001)[ 313] Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon,Morihiko Saito , 2001)[ 314] Modularity theorem (Christophe Breuil ,Brian Conrad ,Fred Diamond , andRichard Taylor , 2001)[ 315] Erdős–Stewart conjecture (Florian Luca , 2001)[ 316] Berry–Robbins problem (Michael Atiyah , 2000)[ 317] ^ A counterexample has been announced, with a preprint made available onarXiv .[ 90] ^ A disproof has been announced, with a preprint made available onarXiv .[ 186] ^ Thiele, Rüdiger (2005). 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Somerville, Massachusetts: International Press. pp. 1– 15.doi :10.4310/SDG.2002.v7.n1.a1 .MR 1919420 .Books discussing problems solved since 1995 [ edit ] Books discussing unsolved problems [ edit ] Chung, Fan ;Graham, Ron (1999).Erdös on Graphs: His Legacy of Unsolved Problems ISBN 978-1-56881-111-6 Croft, Hallard T. ;Falconer, Kenneth J. ;Guy, Richard K. (1994).Unsolved Problems in Geometry 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 ISBN 978-0-88385-315-3 du Sautoy, Marcus (2003).The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics ISBN 978-0-06-093558-0 Derbyshire, John (2003).Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics 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 (inactive 30 January 2026).ISSN 1609-3321 .S2CID 11845578 .Zbl 1066.11030 .{{cite journal }}: CS1 maint: DOI inactive as of January 2026 (link )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 ].
