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List of unsolved problems in mathematics
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<h2 id="lists-of-unsolved-problems-in-mathematics">Lists of unsolved problems in mathematics</h2> <p>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. </p> <table><tbody><tr><th>List</th><th>Number ofproblems</th><th>Number unsolved or incompletely solved</th><th>Proposed by</th><th>Proposedin</th></tr><tr><td><a href="/facts/Hilbert%27s_problems/HaQWIxvV">Hilbert's problems</a><a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></td><td>23</td><td>15</td><td><a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a></td><td>1900</td></tr><tr><td><a href="/facts/Landau%27s_problems/vqLTtmeR">Landau's problems</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></td><td>4</td><td>4</td><td><a href="/facts/Edmund_Landau/lVgFVLBw">Edmund Landau</a></td><td>1912</td></tr><tr><td><a href="/facts/Taniyama%27s_problems/6oqKru8S">Taniyama's problems</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></td><td>36</td><td>–</td><td><a href="/facts/Yutaka_Taniyama/1EGULkp1">Yutaka Taniyama</a></td><td>1955</td></tr><tr><td><a href="/facts/Thurston%27s_24_questions/Nl0upss5">Thurston's 24 questions</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></td><td>24</td><td>–</td><td><a href="/facts/William_Thurston/K1Qe8801">William Thurston</a></td><td>1982</td></tr><tr><td><a href="/facts/Smale%27s_problems/Lep2Fld4">Smale's problems</a></td><td>18</td><td>14</td><td><a href="/facts/Stephen_Smale/vtMg3cAF">Stephen Smale</a></td><td>1998</td></tr><tr><td><a href="/facts/Millennium_Prize_Problems/WWSB5ddp">Millennium Prize Problems</a></td><td>7</td><td>6<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></td><td><a href="/facts/Clay_Mathematics_Institute/9sCIUpD7">Clay Mathematics Institute</a></td><td>2000</td></tr><tr><td><a href="/facts/Simon_problems/ylC91CLP">Simon problems</a></td><td>15</td><td>< 12<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></td><td><a href="/facts/Barry_Simon/BImmcE2I">Barry Simon</a></td><td>2000</td></tr><tr><td>Unsolved Problems on Mathematics for the 21st Century<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></td><td>22</td><td>–</td><td>Jair Minoro Abe, Shotaro Tanaka</td><td>2001</td></tr><tr><td><a href="/facts/DARPA/Uf9sarhT">DARPA</a>'s math challenges<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></td><td>23</td><td>–</td><td><a href="/facts/DARPA/Uf9sarhT">DARPA</a></td><td>2007</td></tr><tr><td>Erdős's problems<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></td><td>> 934</td><td>617</td><td><a href="/facts/Paul_Erd%C5%91s/63LoUTe1">Paul Erdős</a></td><td>Over six decades of Erdős' career, from the 1930s to 1990s</td></tr></tbody></table> <h3>Millennium Prize Problems</h3> <p>Of the original seven <a href="/facts/Millennium_Prize_Problems/WWSB5ddp">Millennium Prize Problems</a> listed by the <a href="/facts/Clay_Mathematics_Institute/9sCIUpD7">Clay Mathematics Institute</a> in 2000, six remain unsolved to date:<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> <ul><li><a href="/facts/Birch_and_Swinnerton-Dyer_conjecture/Issh7Yfr">Birch and Swinnerton-Dyer conjecture</a></li> <li><a href="/facts/Hodge_conjecture/rk31uUHv">Hodge conjecture</a></li> <li><a href="/facts/Navier%E2%80%93Stokes_existence_and_smoothness/5aDcIPCw">Navier–Stokes existence and smoothness</a></li> <li><a href="/facts/P_versus_NP_problem/xoC7Ruda">P versus NP</a></li> <li><a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a></li> <li><a href="/facts/Yang%E2%80%93Mills_existence_and_mass_gap/J27rYXyv">Yang–Mills existence and mass gap</a></li></ul> <p>The seventh problem, the <a href="/facts/Poincar%C3%A9_conjecture/mxk2Pju3">Poincaré conjecture</a>, was solved by <a href="/facts/Grigori_Perelman/SmISWW7x">Grigori Perelman</a> in 2003.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> However, a generalization called the <a href="/facts/Generalized_Poincar%C3%A9_conjecture/9wvLDqTq">smooth four-dimensional Poincaré conjecture</a>—that is, whether a <i>four</i>-dimensional <a href="/facts/Topological_sphere/jywYLNM2">topological sphere</a> can have two or more inequivalent <a href="/facts/Smooth_structure/lfvfhqxr">smooth structures</a>—is unsolved.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <h3>Notebooks</h3> <ul><li>The <a href="/facts/Kourovka,_Sverdlovsk_Oblast/wvTov6hV">Kourovka</a> Notebook (<a href="/facts/Russian_language/ETXrTgHV">Russian</a>: Коуровская тетрадь) is a collection of unsolved problems in <a href="/facts/Group_theory/NfowcI5H">group theory</a>, first published in 1965 and updated many times since.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a></li> <li>The <a href="/facts/Yekaterinburg/PgF4pldC">Sverdlovsk</a> Notebook (<a href="/facts/Russian_language/ETXrTgHV">Russian</a>: Свердловская тетрадь) is a collection of unsolved problems in <a href="/facts/Semigroup_theory/rgSdk2kt">semigroup theory</a>, first published in 1965 and updated every 2 to 4 years since.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li> <li>The <a href="/facts/Dniester/UhEG0beX">Dniester</a> Notebook (<a href="/facts/Russian_language/ETXrTgHV">Russian</a>: Днестровская тетрадь) lists several hundred unsolved problems in algebra, particularly <a href="/facts/Ring_theory/VW6svEFe">ring theory</a> and <a href="/facts/Modulus_(algebraic_number_theory)/vTm4JsAQ">modulus theory</a>.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a></li> <li>The Erlagol Notebook (<a href="/facts/Russian_language/ETXrTgHV">Russian</a>: Эрлагольская тетрадь) lists unsolved problems in algebra and <a href="/facts/Model_theory/vLWVV8sQ">model theory</a>.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a></li></ul> <h2 id="unsolved-problems">Unsolved problems</h2> <h3>Algebra</h3> <p class="note">Main article: <a href="/facts/Algebra/nMdqczjo">Algebra</a></p> <ul><li><a href="/facts/Birch%E2%80%93Tate_conjecture/UjPHPasQ">Birch–Tate conjecture</a> on the relation between the order of the <a href="/facts/Center_(group_theory)/gqAg6B8I">center</a> of the <a href="/facts/Steinberg_group_(K-theory)/MJoD8o5p">Steinberg group</a> of the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> of a <a href="/facts/Algebraic_number_field/e0K0TXSh">number field</a> to the field's <a href="/facts/Dedekind_zeta_function/Nj8PwFo8">Dedekind zeta function</a>.</li> <li><a href="/facts/Bombieri%E2%80%93Lang_conjecture/V2Jd3mqP">Bombieri–Lang conjectures</a> on densities of rational points of <a href="/facts/Algebraic_surface/2zHU0o9J">algebraic surfaces</a> and <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic varieties</a> defined on <a href="/facts/Algebraic_number_field/e0K0TXSh">number fields</a> and their <a href="/facts/Field_extension/L7yIDtyK">field extensions</a>.</li> <li><a href="/facts/Connes_embedding_problem/UEnlnbS3">Connes embedding problem</a> in <a href="/facts/Von_Neumann_algebra/8NyWUIRH">Von Neumann algebra</a> theory</li> <li><a href="/facts/Crouzeix%27s_conjecture/mdakXHgz">Crouzeix's conjecture</a>: the <a href="/facts/Matrix_norm/UJvsb9fF">matrix norm</a> of a complex function f {\displaystyle f} applied to a complex matrix A {\displaystyle A} is at most twice the <a href="/facts/Infimum_and_supremum/oXCKVopz">supremum</a> of | f ( z ) | {\displaystyle |f(z)|} over the <a href="/facts/Numerical_range/4gCh6ShW">field of values</a> of A {\displaystyle A} .</li> <li><a href="/facts/Determinantal_conjecture/GvT63taw">Determinantal conjecture</a> on the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the sum of two <a href="/facts/Normal_matrix/WVKxrEK9">normal matrices</a>.</li> <li><a href="/facts/Eilenberg%E2%80%93Ganea_conjecture/Ip3Ez1Rm">Eilenberg–Ganea conjecture</a>: a group with <a href="/facts/Cohomological_dimension/m9jYzNAB">cohomological dimension</a> 2 also has a 2-dimensional <a href="/facts/Eilenberg%E2%80%93MacLane_space/mrux6oMJ">Eilenberg–MacLane space</a> K ( G , 1 ) {\displaystyle K(G,1)} .</li> <li><a href="/facts/Farrell%E2%80%93Jones_conjecture/VSt9xdER">Farrell–Jones conjecture</a> on whether certain <a href="/facts/Assembly_map/0KfeaLg4">assembly maps</a> are <a href="/facts/Isomorphisms/pj8KgkxU">isomorphisms</a>. <ul><li><a href="/facts/Farrell%E2%80%93Jones_conjecture/VSt9xdER">Bost conjecture</a>: a specific case of the Farrell–Jones conjecture</li></ul></li> <li><a href="/facts/Finite_lattice_representation_problem/GH0fA2Tt">Finite lattice representation problem</a>: is every finite <a href="/facts/Lattice_(order)/5ak6ufky">lattice</a> isomorphic to the <a href="/facts/Quotient_(universal_algebra)/8uQzYkm7">congruence lattice</a> of some finite <a href="/facts/Universal_algebra/uMJAL6ER">algebra</a>?<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a></li> <li><a href="/facts/Goncharov_conjecture/0jER1QID">Goncharov conjecture</a> on the <a href="/facts/Cohomology/J7FJzTNh">cohomology</a> of certain <a href="/facts/Motivic_cohomology/l6XUznW4">motivic complexes</a>.</li> <li><a href="/facts/Green%27s_conjecture/XuPqgRbt">Green's conjecture</a>: the <a href="/facts/Clifford%27s_theorem_on_special_divisors/XuPqgRbt">Clifford index</a> of a non-<a href="/facts/Hyperelliptic_curve/sf07gwGS">hyperelliptic curve</a> is determined by the extent to which it, as a <a href="/facts/Canonical_bundle/nJKN8mpG">canonical curve</a>, has <a href="/facts/Linear_relation/QQYSD9I8">linear syzygies</a>.</li> <li><a href="/facts/Grothendieck%E2%80%93Katz_p-curvature_conjecture/psdXPfrM">Grothendieck–Katz p-curvature conjecture</a>: a conjectured <a href="/facts/Hasse_principle/B6el223t">local–global principle</a> for <a href="/facts/Linear_differential_equation/nLEbIIVf">linear ordinary differential equations</a>.</li> <li><a href="/facts/Hadamard_conjecture/q56o9lgM">Hadamard conjecture</a>: for every positive integer k {\displaystyle k} , a <a href="/facts/Hadamard_matrix/q56o9lgM">Hadamard matrix</a> of order 4 k {\displaystyle 4k} exists. <ul><li><a href="/facts/Williamson_conjecture/yn1A4QEm">Williamson conjecture</a>: the problem of finding Williamson matrices, which can be used to construct Hadamard matrices.</li></ul></li> <li><a href="/facts/Hadamard%27s_maximal_determinant_problem/hAz1JDsE">Hadamard's maximal determinant problem</a>: what is the largest <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of a matrix with entries all equal to 1 or –1?</li> <li><a href="/facts/Hilbert%27s_fifteenth_problem/aw8mKcfE">Hilbert's fifteenth problem</a>: put <a href="/facts/Schubert_calculus/1U1ayXqR">Schubert calculus</a> on a rigorous foundation.</li> <li><a href="/facts/Hilbert%27s_sixteenth_problem/SUR6JBKB">Hilbert's sixteenth problem</a>: what are the possible configurations of the <a href="/facts/Connected_space/5CUTxfoA">connected components</a> of <a href="/facts/Harnack%27s_curve_theorem/I2s9njop">M-curves</a>?</li> <li><a href="/facts/Homological_conjectures_in_commutative_algebra/ndl1pLef">Homological conjectures in commutative algebra</a></li> <li><a href="/facts/Jacobson%27s_conjecture/rCfjaWqn">Jacobson's conjecture</a>: the intersection of all powers of the <a href="/facts/Jacobson_radical/tTTfKySl">Jacobson radical</a> of a left-and-right <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian ring</a> is precisely 0.</li> <li><a href="/facts/Kaplansky%27s_conjectures/uP0oXvJf">Kaplansky's conjectures</a></li> <li><a href="/facts/K%C3%B6the_conjecture/DMQxAYpw">Köthe conjecture</a>: if a ring has no <a href="/facts/Nil_ideal/KDhmC0uc">nil ideal</a> other than { 0 } {\displaystyle \{0\}} , then it has no nil <a href="/facts/Ideal_(ring_theory)/vCvytduX">one-sided ideal</a> other than { 0 } {\displaystyle \{0\}} .</li> <li><a href="/facts/Monomial_conjecture/J7MzDxkN">Monomial conjecture</a> on <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a> <a href="/facts/Local_ring/gRTdg5Qx">local rings</a></li> <li>Existence of <a href="/facts/Perfect_cuboid/9h5MIJrK">perfect cuboids</a> and associated <a href="/facts/Cuboid_conjectures/9h5MIJrK">cuboid conjectures</a></li> <li><a href="/facts/Pierce%E2%80%93Birkhoff_conjecture/VzCY8nBU">Pierce–Birkhoff conjecture</a>: every piecewise-polynomial f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is the maximum of a finite set of minimums of finite collections of polynomials.</li> <li><a href="/facts/Rota%27s_basis_conjecture/81megqL5">Rota's basis conjecture</a>: for matroids of rank n {\displaystyle n} with n {\displaystyle n} disjoint bases B i {\displaystyle B_{i}} , it is possible to create an n × n {\displaystyle n\times n} matrix whose rows are B i {\displaystyle B_{i}} and whose columns are also bases.</li> <li><a href="/facts/Serre%27s_conjecture_II_(algebra)/yrRD1e62">Serre's conjecture II</a>: if G {\displaystyle G} is a <a href="/facts/Simply_connected_space/xFlta63J">simply connected</a> <a href="/facts/Semisimple_algebraic_group/z1zyJvUh">semisimple algebraic group</a> over a perfect <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Cohomological_dimension/m9jYzNAB">cohomological dimension</a> at most 2 {\displaystyle 2} , then the <a href="/facts/Galois_cohomology/nzC1EVYT">Galois cohomology</a> set H 1 ( F , G ) {\displaystyle H^{1}(F,G)} is zero.</li> <li><a href="/facts/Serre%27s_multiplicity_conjectures/lIA8APPD">Serre's positivity conjecture</a> that if R {\displaystyle R} is a commutative <a href="/facts/Regular_local_ring/P9s5Fvrl">regular local ring</a>, and P , Q {\displaystyle P,Q} are <a href="/facts/Prime_ideal/pmrGf6DA">prime ideals</a> of R {\displaystyle R} , then dim ( R / P ) + dim ( R / Q ) = dim ( R ) {\displaystyle \dim(R/P)+\dim(R/Q)=\dim(R)} implies χ ( R / P , R / Q ) > 0 {\displaystyle \chi (R/P,R/Q)>0} .</li> <li><a href="/facts/Uniform_boundedness_conjecture_for_rational_points/74tLlRwE">Uniform boundedness conjecture for rational points</a>: do <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curves</a> of <a href="/facts/Geometric_genus/S2oQVNCb">genus</a> g ≥ 2 {\displaystyle g\geq 2} over <a href="/facts/Algebraic_number_field/e0K0TXSh">number fields</a> K {\displaystyle K} have at most some bounded number N ( K , g ) {\displaystyle N(K,g)} of K {\displaystyle K} -<a href="/facts/Rational_point/j5v4cxuE">rational points</a>?</li> <li><a href="/facts/Wild_problem/Vsmz8s1J">Wild problems</a>: problems involving classification of pairs of n × n {\displaystyle n\times n} matrices under simultaneous conjugation.</li> <li><a href="/facts/Zariski%E2%80%93Lipman_conjecture/PyI6P5gF">Zariski–Lipman conjecture</a>: for a <a href="/facts/Complex_algebraic_variety/4dTr8zfR">complex algebraic variety</a> V {\displaystyle V} with <a href="/facts/Coordinate_ring/WVynz2Kn">coordinate ring</a> R {\displaystyle R} , if the <a href="/facts/Derivation_(algebra)/vcSqpxzu">derivations</a> of R {\displaystyle R} are a <a href="/facts/Free_module/o6rC6yoJ">free module</a> over R {\displaystyle R} , then V {\displaystyle V} is <a href="/facts/Smooth_algebraic_variety/8nvkrkE9">smooth</a>.</li> <li>Zauner's conjecture: do <a href="/facts/SIC-POVM/qqfeUfCA">SIC-POVMs</a> exist in all dimensions?</li> <li><a href="/facts/Zilber%E2%80%93Pink_conjecture/d66kGcF5">Zilber–Pink conjecture</a> that if X {\displaystyle X} is a mixed <a href="/facts/Shimura_variety/varFgmR6">Shimura variety</a> or <a href="/facts/Abelian_variety/dGSDVruG">semiabelian variety</a> defined over C {\displaystyle \mathbb {C} } , and V ⊆ X {\displaystyle V\subseteq X} is a subvariety, then V {\displaystyle V} contains only finitely many atypical subvarieties.</li></ul> <h4>Group theory</h4> <p class="note">Main article: <a href="/facts/Group_theory/NfowcI5H">Group theory</a></p> <ul><li><a href="/facts/Andrews%E2%80%93Curtis_conjecture/tR6TDK2L">Andrews–Curtis conjecture</a>: every balanced <a href="/facts/Presentation_of_a_group/f9emPUoI">presentation</a> of the <a href="/facts/Trivial_group/XPjQkQFm">trivial group</a> can be transformed into a trivial presentation by a sequence of <a href="/facts/Nielsen_transformation/tx5VeX06">Nielsen transformations</a> on <a href="/facts/Presentation_of_a_group/f9emPUoI">relators</a> and conjugations of relators</li> <li><a href="/facts/Burnside_problem/RsWB8d31">Bounded Burnside problem</a>: for which positive integers <i>m</i>, <i>n</i> is the free Burnside group B(<i>m</i>,<i>n</i>) finite? In particular, is B(2, 5) finite?</li> <li>Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a></li> <li><a href="/facts/Herzog%E2%80%93Sch%C3%B6nheim_conjecture/tDjAnPrW">Herzog–Schönheim conjecture</a>: if a finite system of left <a href="/facts/Coset/zFgQUTLY">cosets</a> of subgroups of a group G {\displaystyle G} form a partition of G {\displaystyle G} , then the finite indices of said subgroups cannot be distinct.</li> <li>The <a href="/facts/Inverse_Galois_problem/phfoWPLl">inverse Galois problem</a>: is every finite group the Galois group of a Galois extension of the rationals?</li> <li><a href="/facts/Isomorphism_problem_of_Coxeter_groups/otIUlSH8">Isomorphism problem of Coxeter groups</a></li> <li>Are there an infinite number of <a href="/facts/Leinster_group/YJy6eclk">Leinster groups</a>?</li> <li>Does <a href="/facts/Monstrous_moonshine/WbyFQ68w">generalized moonshine</a> exist?</li> <li>Is every <a href="/facts/Finitely_presented_group/f9emPUoI">finitely presented</a> <a href="/facts/Periodic_group/TEvLhDtm">periodic group</a> finite?</li> <li>Is every group <a href="/facts/Surjunctive_group/zgD8JbTd">surjunctive</a>?</li> <li>Is every discrete, countable group <a href="/facts/Sofic_group/x1FtRzcc">sofic</a>?</li> <li><a href="/facts/Problems_in_loop_theory_and_quasigroup_theory/smUvKsEE">Problems in loop theory and quasigroup theory</a> consider generalizations of groups</li></ul> <h4>Representation theory</h4> <ul><li><a href="/facts/Arthur%27s_conjectures/QI9q4Tuo">Arthur's conjectures</a></li> <li><a href="/facts/Dade%27s_conjecture/T1pKwvtv">Dade's conjecture</a> relating the numbers of <a href="/facts/Character_theory/WMp5Q6YE">characters</a> of <a href="/facts/Modular_representation_theory/2FIf7j4j">blocks</a> of a finite group to the numbers of characters of blocks of local <a href="/facts/Subgroup/r91mBgpa">subgroups</a>.</li> <li><a href="/facts/Demazure_conjecture/Jl1Xho3m">Demazure conjecture</a> on <a href="/facts/Group_representation/FsuuDagQ">representations</a> of <a href="/facts/Algebraic_group/DDtBCIvh">algebraic groups</a> over the integers.</li> <li><a href="/facts/Kazhdan%E2%80%93Lusztig_polynomial/kC0JqxUM">Kazhdan–Lusztig conjectures</a> relating the values of the <a href="/facts/Kazhdan%E2%80%93Lusztig_polynomial/kC0JqxUM">Kazhdan–Lusztig polynomials</a> at 1 with <a href="/facts/Group_representation/FsuuDagQ">representations</a> of complex <a href="/facts/Semisimple_Lie_algebra/m8jA3Bcf">semisimple Lie groups</a> and <a href="/facts/Semisimple_Lie_algebra/m8jA3Bcf">Lie algebras</a>.</li> <li><a href="/facts/McKay_conjecture/tF9dWITF">McKay conjecture</a>: in a group G {\displaystyle G} , the number of <a href="/facts/Character_theory/WMp5Q6YE">irreducible complex characters</a> of degree not divisible by a <a href="/facts/Prime_number/L20FLkgn">prime number</a> p {\displaystyle p} is equal to the number of irreducible complex characters of the <a href="/facts/Centralizer_and_normalizer/AKhrEUGR">normalizer</a> of any <a href="/facts/Sylow_theorems/aVTxDRUo">Sylow p {\displaystyle p} -subgroup</a> within G {\displaystyle G} .</li></ul> <h3>Analysis</h3> <p class="note">Main article: <a href="/facts/Mathematical_analysis/XXeR26Ih">Mathematical analysis</a></p> <ul><li>The <a href="/facts/Brennan_conjecture/z5aKFbbX">Brennan conjecture</a>: estimating the integral of powers of the moduli of the derivative of <a href="/facts/Conformal_map/cEpQnMad">conformal maps</a> into the open unit disk, on certain subsets of C {\displaystyle \mathbb {C} } </li> <li><a href="/facts/Fuglede%27s_conjecture/SqnH4Inh">Fuglede's conjecture</a> on whether nonconvex sets in R {\displaystyle \mathbb {R} } and R 2 {\displaystyle \mathbb {R} ^{2}} are spectral if and only if they tile by <a href="/facts/Translation_(geometry)/KlpWmnfX">translation</a>.</li> <li>Goodman's conjecture on the coefficients of <a href="/facts/Multivalent_function/WH5ua6BL">multivalued functions</a></li> <li><a href="/facts/Invariant_subspace_problem/IHQF0rCc">Invariant subspace problem</a> – does every <a href="/facts/Bounded_operator/LYmDTIqR">bounded operator</a> on a complex <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> send some non-trivial <a href="/facts/Closed_set/upYnRTUj">closed</a> subspace to itself?</li> <li>Kung–Traub conjecture on the optimal order of a multipoint iteration without memory<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a></li> <li><a href="/facts/Lehmer%27s_conjecture/rkqsVlwy">Lehmer's conjecture</a> on the Mahler measure of non-cyclotomic polynomials<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a></li> <li>The <a href="/facts/Mean_value_problem/z5uLyTgN">mean value problem</a>: given a <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> f {\displaystyle f} of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> d ≥ 2 {\displaystyle d\geq 2} and a complex number z {\displaystyle z} , is there a <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical point</a> c {\displaystyle c} of f {\displaystyle f} such that | f ( z ) − f ( c ) | ≤ | f ′ ( z ) | | z − c | {\displaystyle |f(z)-f(c)|\leq |f'(z)||z-c|} ?</li> <li>The <a href="/facts/Pompeiu_problem/mnTC4xvF">Pompeiu problem</a> on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a></li> <li><a href="/facts/Sendov%27s_conjecture/7vPfgLUK">Sendov's conjecture</a>: if a complex polynomial with degree at least 2 {\displaystyle 2} has all roots in the closed <a href="/facts/Unit_disk/phrUu7yC">unit disk</a>, then each root is within distance 1 {\displaystyle 1} from some <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical point</a>.</li> <li><a href="/facts/Analytic_capacity/CWAoWnlt">Vitushkin's conjecture</a> on compact subsets of C {\displaystyle \mathbb {C} } with <a href="/facts/Analytic_capacity/CWAoWnlt">analytic capacity</a> 0 {\displaystyle 0} </li> <li>What is the exact value of <a href="/facts/Landau%27s_constants/67MQUFAt">Landau's constants</a>, including <a href="/facts/Bloch%27s_theorem_(complex_variables)/Y2ok21No">Bloch's constant</a>?</li></ul> <ul><li>Regularity of solutions of <a href="/facts/Euler_equations_(fluid_dynamics)/EYW5lerl">Euler equations</a></li> <li>Convergence of <a href="https://mathworld.wolfram.com/FlintHillsSeries.html">Flint Hills series</a></li> <li>Regularity of solutions of <a href="/facts/Vlasov%E2%80%93Maxwell_equations/hrNKvMp3">Vlasov–Maxwell equations</a></li></ul> <h3>Combinatorics</h3> <p class="note">Main article: <a href="/facts/Combinatorics/k14xUoKT">Combinatorics</a></p> <ul><li>The <a href="/facts/1/3%E2%80%932/3_conjecture/E3INzB0D">1/3–2/3 conjecture</a> – does every finite <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> that is not <a href="/facts/Totally_ordered/xnTtmuYJ">totally ordered</a> contain two elements <i>x</i> and <i>y</i> such that the probability that <i>x</i> appears before <i>y</i> in a random <a href="/facts/Linear_extension/jKKPR2pn">linear extension</a> is between 1/3 and 2/3?<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a></li> <li>The <a href="/facts/Dittert_conjecture/TFrwt9VH">Dittert conjecture</a> concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition</li> <li><a href="/facts/Problems_in_Latin_squares/uqoMpvgU">Problems in Latin squares</a> – open questions concerning <a href="/facts/Latin_squares/1e60MInk">Latin squares</a></li> <li>The <a href="/facts/Lonely_runner_conjecture/os5KxC2n">lonely runner conjecture</a> – if k {\displaystyle k} runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance 1 / k {\displaystyle 1/k} from each other runner) at some time?<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a></li> <li><a href="/facts/Map_folding/WyzwN7TH">Map folding</a> – various problems in map folding and stamp folding.</li> <li><a href="/facts/No-three-in-line_problem/HVhCJN0c">No-three-in-line problem</a> – how many points can be placed in the n × n {\displaystyle n\times n} grid so that no three of them lie on a line?</li> <li><a href="/facts/Rudin%27s_conjecture/mB9o5gQm">Rudin's conjecture</a> on the number of squares in finite <a href="/facts/Arithmetic_progression/c4ew9Xoq">arithmetic progressions</a><a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a></li> <li>The <a href="/facts/Sunflower_(mathematics)/YNHlPlet">sunflower conjecture</a> – can the number of k {\displaystyle k} size sets required for the existence of a sunflower of r {\displaystyle r} sets be bounded by an exponential function in k {\displaystyle k} for every fixed r > 2 {\displaystyle r>2} ?</li> <li>Frankl's <a href="/facts/Union-closed_sets_conjecture/QLWVxCCF">union-closed sets conjecture</a> – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a></li></ul> <ul><li>Give a combinatorial interpretation of the <a href="/facts/Kronecker_coefficient/mt2krLNx">Kronecker coefficients</a><a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a></li> <li>The values of the <a href="/facts/Dedekind_number/vnzw9GrY">Dedekind numbers</a> M ( n ) {\displaystyle M(n)} for n ≥ 10 {\displaystyle n\geq 10} <a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a></li> <li>The values of the <a href="/facts/Ramsey_numbers/VS6K8jGk">Ramsey numbers</a>, particularly R ( 5 , 5 ) {\displaystyle R(5,5)} </li> <li>The values of the <a href="/facts/Van_der_Waerden_number/e0Lsv0zr">Van der Waerden numbers</a></li> <li>Finding a function to model n-step <a href="/facts/Self-avoiding_walk/23chozBY">self-avoiding walks</a><a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a></li></ul> <h3>Dynamical systems</h3> <p class="note">Main article: <a href="/facts/Dynamical_system/782gREl5">Dynamical system</a></p> <ul><li><a href="/facts/Arnold%E2%80%93Givental_conjecture/XMyysC3y">Arnold–Givental conjecture</a> and <a href="/facts/Symplectomorphism/M6nBcau4">Arnold conjecture</a> – relating symplectic geometry to Morse theory.</li> <li><a href="/facts/Quantum_chaos/p1JVxSTD">Berry–Tabor conjecture</a> in <a href="/facts/Quantum_chaos/p1JVxSTD">quantum chaos</a></li> <li><a href="/facts/Stefan_Banach/b1qHSNg2">Banach's</a> problem – is there an <a href="/facts/Measure-preserving_dynamical_system/acIdlpqt">ergodic system</a> with simple Lebesgue spectrum?<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a></li> <li><a href="/facts/George_David_Birkhoff/3aywz7gs">Birkhoff</a> conjecture – if a <a href="/facts/Dynamical_billiards/2HPqXIrS">billiard table</a> is strictly convex and integrable, is its boundary necessarily an ellipse?<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a></li> <li><a href="/facts/Collatz_conjecture/nr3K8p4n">Collatz conjecture</a> (also known as the 3 n + 1 {\displaystyle 3n+1} conjecture)</li> <li><a href="/facts/Eden%27s_conjecture/9sDVXfXf">Eden's conjecture</a> that the <a href="/facts/Infimum_and_supremum/oXCKVopz">supremum</a> of the local <a href="/facts/Lyapunov_dimension/XmDv18fM">Lyapunov dimensions</a> on the global <a href="/facts/Attractor/FAx5Db4w">attractor</a> is achieved on a stationary point or an unstable periodic orbit embedded into the attractor.</li> <li><a href="/facts/Alexandre_Eremenko/ohfIII8t">Eremenko's</a> conjecture: every component of the <a href="/facts/Escaping_set/eEYsSzH4">escaping set</a> of an <a href="/facts/Entire_function/XNluKzu7">entire</a> <a href="/facts/Transcendental_function/yMkeq4Eo">transcendental</a> function is unbounded.</li> <li><a href="/facts/Fatou_conjecture/grLXb69X">Fatou conjecture</a> that a quadratic family of maps from the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> to itself is hyperbolic for an open dense set of parameters.</li> <li><a href="/facts/Hillel_Furstenberg/keLWMrKw">Furstenberg</a> conjecture – is every invariant and <a href="/facts/Ergodicity/CKY3Gr9v">ergodic</a> measure for the × 2 , × 3 {\displaystyle \times 2,\times 3} action on the circle either Lebesgue or atomic?</li> <li><a href="/facts/Kaplan%E2%80%93Yorke_conjecture/pltdwU1B">Kaplan–Yorke conjecture</a> on the dimension of an <a href="/facts/Attractor/FAx5Db4w">attractor</a> in terms of its <a href="/facts/Lyapunov_exponent/VJDB5K5T">Lyapunov exponents</a></li> <li><a href="/facts/Grigory_Margulis/kTkcHJII">Margulis</a> conjecture – measure classification for diagonalizable actions in higher-rank groups.</li> <li><a href="/facts/Hilbert%E2%80%93Arnold_problem/Qruv3KDl">Hilbert–Arnold problem</a> – is there a <a href="/facts/Uniformly_bounded/nyxwk4Ix">uniform bound</a> on <a href="/facts/Limit_cycles/LmM9EVoA">limit cycles</a> in generic finite-parameter families of <a href="/facts/Vector_fields/ccFNuPPp">vector fields</a> on a sphere?</li> <li><a href="/facts/MLC_conjecture/2NVUgrVG">MLC conjecture</a> – is the Mandelbrot set locally connected?</li> <li>Many problems concerning an <a href="/facts/Outer_billiard/VgsVtUzX">outer billiard</a>, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.</li> <li>Quantum unique ergodicity conjecture on the distribution of large-frequency <a href="/facts/Eigenfunction/62EJS0XB">eigenfunctions</a> of the <a href="/facts/Laplace_operator/kov9u3PT">Laplacian</a> on a <a href="/facts/Curvature/RFCAEsj8">negatively-curved</a> <a href="/facts/Manifold/rXPERJtu">manifold</a><a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a></li> <li><a href="/facts/Vladimir_Abramovich_Rokhlin/QleBUYCI">Rokhlin's</a> multiple mixing problem – are all <a href="/facts/Mixing_(mathematics)/ttsAJCID">strongly mixing</a> systems also strongly 3-mixing?<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a></li> <li><a href="/facts/Weinstein_conjecture/ERJ6B0jN">Weinstein conjecture</a> – does a regular compact <a href="/facts/Contact_type/YwGA7Sax">contact type</a> <a href="/facts/Level_set/JUmNbdNh">level set</a> of a <a href="/facts/Hamiltonian_function/2ReerlNA">Hamiltonian</a> on a <a href="/facts/Symplectic_manifold/JvNrJkC7">symplectic manifold</a> carry at least one periodic orbit of the Hamiltonian flow?</li></ul> <ul><li>Does every positive integer generate a <a href="/facts/Juggler_sequence/nhQW03cX">juggler sequence</a> terminating at 1?</li> <li><a href="/facts/Lyapunov_function/RB590Utb">Lyapunov function: Lyapunov's second method for stability</a> – For what classes of <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ODEs</a>, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?</li> <li>Is every <a href="/facts/Reversible_cellular_automaton/NE63q0pq">reversible cellular automaton</a> in three or more dimensions locally reversible?<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a></li></ul> <h3>Games and puzzles</h3> <p class="note">Main article: <a href="/facts/Game_theory/zkEIj1ya">Game theory</a></p> <h4>Combinatorial games</h4> <p class="note">Main article: <a href="/facts/Combinatorial_game_theory/fF4l0om2">Combinatorial game theory</a></p> <ul><li><a href="/facts/Sudoku/I5Savmhf">Sudoku</a>: <ul><li>How many puzzles have exactly one solution?<a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a></li> <li>How many puzzles with exactly one solution are <a href="/facts/Glossary_of_Sudoku/9XNxdCMH">minimal</a>?<a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a></li> <li>What is the <a href="/facts/Mathematics_of_Sudoku/fcJqLklG">maximum number of givens</a> for a <a href="/facts/Glossary_of_Sudoku/9XNxdCMH">minimal</a> puzzle?<a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a></li></ul></li> <li><a href="/facts/Tic-tac-toe_variants/XhUfnkFA">Tic-tac-toe variants</a>: <ul><li>Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also <a href="/facts/Hales%E2%80%93Jewett_theorem/53eK8f7x">Hales–Jewett theorem</a> and <a href="/facts/Nd_game/M3SBjtWA"><i>n</i><i>d</i> game</a>)<a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a></li></ul></li> <li><a href="/facts/Chess/47ykvkHF">Chess</a>: <ul><li>What is the outcome of a perfectly played game of chess? (See also <a href="/facts/First-move_advantage_in_chess/mKwI68o8">first-move advantage in chess</a>)</li></ul></li> <li><a href="/facts/Go_(game)/1fKhemww">Go</a>: <ul><li>What is the perfect value of <a href="/facts/Komi_(Go)/NffhtWcE">Komi</a>?</li></ul></li> <li>Are the nim-sequences of all finite <a href="/facts/Octal_game/saPHqtH9">octal games</a> eventually periodic?</li> <li>Is the nim-sequence of <a href="/facts/Grundy%27s_game/cv0bllSu">Grundy's game</a> eventually periodic?</li></ul> <h4>Games with imperfect information</h4> <ul><li><a href="/facts/Rendezvous_problem/xPLF3Qna">Rendezvous problem</a></li></ul> <h3>Geometry</h3> <p class="note">Main article: <a href="/facts/Geometry/wTQf5ffQ">Geometry</a></p> <h4>Algebraic geometry</h4> <p class="note">Main article: <a href="/facts/Algebraic_geometry/rh7RHVp3">Algebraic geometry</a></p> <ul><li><a href="/facts/Abundance_conjecture/9cSAtI7J">Abundance conjecture</a>: if the <a href="/facts/Canonical_bundle/nJKN8mpG">canonical bundle</a> of a <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a> with <a href="/facts/Canonical_singularity/cCbkz2NR">Kawamata log terminal singularities</a> is <a href="/facts/Nef_line_bundle/TdTxFsKC">nef</a>, then it is semiample.</li> <li><a href="/facts/Bass_conjecture/CEvMSDWg">Bass conjecture</a> on the <a href="/facts/Finitely_generated_group/fVsFHrmb">finite generation</a> of certain <a href="/facts/Algebraic_K-theory/0RY5M49W">algebraic K-groups</a>.</li> <li><a href="/facts/Bass%E2%80%93Quillen_conjecture/7PdUlLfa">Bass–Quillen conjecture</a> relating <a href="/facts/Vector_bundle/cXACAa1I">vector bundles</a> over a <a href="/facts/Regular_local_ring/P9s5Fvrl">regular</a> <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian ring</a> and over the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> A [ t 1 , … , t n ] {\displaystyle A[t_{1},\ldots ,t_{n}]} .</li> <li><a href="/facts/Deligne_conjecture/wYC8fhfq">Deligne conjecture</a>: any one of numerous named for <a href="/facts/Pierre_Deligne/wYC8fhfq">Pierre Deligne</a>. <ul><li><a href="/facts/Deligne%27s_conjecture_on_Hochschild_cohomology/b8MP4Yp2">Deligne's conjecture on Hochschild cohomology</a> about the <a href="/facts/Operad/RfUyH38R">operadic</a> structure on <a href="/facts/Hochschild_homology/0phwDIyX">Hochschild cochain complex</a>.</li></ul></li> <li><a href="/facts/Dixmier_conjecture/PL1q1AxU">Dixmier conjecture</a>: any <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a> of a <a href="/facts/Weyl_algebra/PADSD0k9">Weyl algebra</a> is an <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a>.</li> <li><a href="/facts/Fr%C3%B6berg_conjecture/Xx7ccwA9">Fröberg conjecture</a> on the <a href="/facts/Hilbert_series_and_Hilbert_polynomial/FtFu54gb">Hilbert functions</a> of a set of forms.</li> <li><a href="/facts/Fujita_conjecture/s1BGJaq8">Fujita conjecture</a> regarding the line bundle K M ⊗ L ⊗ m {\displaystyle K_{M}\otimes L^{\otimes m}} constructed from a <a href="/facts/Positive_form/EPFU8KEC">positive</a> <a href="/facts/Holomorphic_vector_bundle/FeRrK9Bg">holomorphic line bundle</a> L {\displaystyle L} on a <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Complex_manifold/DlvPnpzm">complex manifold</a> M {\displaystyle M} and the <a href="/facts/Canonical_bundle/nJKN8mpG">canonical line bundle</a> K M {\displaystyle K_{M}} of M {\displaystyle M} </li> <li><a href="/facts/General_elephant/30FtzKfJ">General elephant problem</a>: do <a href="/facts/General_elephant/30FtzKfJ">general elephants</a> have at most <a href="/facts/Du_Val_singularity/pLmdkbGv">Du Val singularities</a>?</li> <li>Hartshorne's conjectures<a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a></li> <li>In <a href="/facts/Spherical_geometry/WGLQb1Ys">spherical</a> or <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a>, must polyhedra with the same volume and <a href="/facts/Dehn_invariant/eahMFAAb">Dehn invariant</a> be <a href="/facts/Scissors-congruent/UcS6SWVR">scissors-congruent</a>?<a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a></li> <li><a href="/facts/Jacobian_conjecture/J2NbRNk4">Jacobian conjecture</a>: if a <a href="/facts/Polynomial_mapping/MqaE4DMA">polynomial mapping</a> over a <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a>-0 field has a constant nonzero <a href="/facts/Jacobian_matrix_and_determinant/0tNv6o8j">Jacobian determinant</a>, then it has a <a href="/facts/Morphism_of_algebraic_varieties/5jBzIgps">regular</a> (i.e. with polynomial components) inverse function.</li> <li><a href="/facts/Manin_conjecture/pkDHXN1H">Manin conjecture</a> on the distribution of <a href="/facts/Rational_point/j5v4cxuE">rational points</a> of bounded <a href="/facts/Height_function/ByeW10AJ">height</a> in certain subsets of <a href="/facts/Fano_variety/or8RszIv">Fano varieties</a></li> <li><a href="/facts/Maulik%E2%80%93Nekrasov%E2%80%93Okounkov%E2%80%93Pandharipande_conjecture/ehRqTTg0">Maulik–Nekrasov–Okounkov–Pandharipande conjecture</a> on an equivalence between <a href="/facts/Gromov%E2%80%93Witten_invariant/EJwrClYB">Gromov–Witten theory</a> and <a href="/facts/Donaldson%E2%80%93Thomas_theory/ehRqTTg0">Donaldson–Thomas theory</a><a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a></li> <li><a href="/facts/Nagata%27s_conjecture_on_curves/jgyzTJeF">Nagata's conjecture on curves</a>, specifically the minimal degree required for a <a href="/facts/Algebraic_curve/LnWsPmDP">plane algebraic curve</a> to pass through a collection of very general points with prescribed <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicities</a>.</li> <li><a href="/facts/Nagata%E2%80%93Biran_conjecture/heAhNuUa">Nagata–Biran conjecture</a> that if X {\displaystyle X} is a smooth <a href="/facts/Algebraic_surface/2zHU0o9J">algebraic surface</a> and L {\displaystyle L} is an <a href="/facts/Ample_line_bundle/XRQKvXGD">ample line bundle</a> on X {\displaystyle X} of degree d {\displaystyle d} , then for sufficiently large r {\displaystyle r} , the <a href="/facts/Seshadri_constant/h22lBMP0">Seshadri constant</a> satisfies ε ( p 1 , … , p r ; X , L ) = d / r {\displaystyle \varepsilon (p_{1},\ldots ,p_{r};X,L)=d/{\sqrt {r}}} .</li> <li><a href="/facts/Nakai_conjecture/PyI6P5gF">Nakai conjecture</a>: if a <a href="/facts/Complex_algebraic_variety/4dTr8zfR">complex algebraic variety</a> has a ring of <a href="/facts/Differential_operator/Nr15J0IE">differential operators</a> generated by its contained <a href="/facts/Derivation_(differential_algebra)/vcSqpxzu">derivations</a>, then it must be <a href="/facts/Singular_point_of_an_algebraic_variety/8nvkrkE9">smooth</a>.</li> <li><a href="/facts/Parshin%27s_conjecture/5QCd0Ns8">Parshin's conjecture</a>: the higher <a href="/facts/Algebraic_K-theory/0RY5M49W">algebraic K-groups</a> of any <a href="/facts/Smooth_morphism/GNxTBHTG">smooth</a> <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a> defined over a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> must vanish up to torsion.</li> <li><a href="/facts/Section_conjecture/9WbSkDJN">Section conjecture</a> on splittings of <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphisms</a> from <a href="/facts/Fundamental_group/VyYtLqSP">fundamental groups</a> of complete <a href="/facts/Curve/xBQ0pTBW">smooth curves</a> over finitely-generated <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a> k {\displaystyle k} to the <a href="/facts/Galois_group/A0p35p5c">Galois group</a> of k {\displaystyle k} .</li> <li><a href="/facts/Standard_conjectures/54xlzvXF">Standard conjectures</a> on algebraic cycles</li> <li><a href="/facts/Tate_conjecture/LCXv2cMw">Tate conjecture</a> on the connection between <a href="/facts/Algebraic_cycle/eY1eYyk3">algebraic cycles</a> on <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic varieties</a> and <a href="/facts/Galois_module/fYDgqs8d">Galois representations</a> on <a href="/facts/%C3%89tale_cohomology/KXJ8a6Hw">étale cohomology groups</a>.</li> <li><a href="/facts/Virasoro_conjecture/8WBhrys9">Virasoro conjecture</a>: a certain <a href="/facts/Generating_function/K7UVjnjL">generating function</a> encoding the <a href="/facts/Gromov%E2%80%93Witten_invariant/EJwrClYB">Gromov–Witten invariants</a> of a <a href="/facts/Singular_point_of_an_algebraic_variety/8nvkrkE9">smooth</a> <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a> is fixed by an action of half of the <a href="/facts/Virasoro_algebra/p84kseUl">Virasoro algebra</a>.</li> <li>Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of <a href="/facts/Algebraic_variety/Vk7f97vn">varieties</a> at <a href="/facts/Singular_point_of_an_algebraic_variety/8nvkrkE9">singular points</a><a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a></li></ul> <ul><li>Are infinite sequences of <a href="/facts/Flip_(mathematics)/a9z0V5Y0">flips</a> possible in dimensions greater than 3?</li> <li><a href="/facts/Resolution_of_singularities/8n4Vj4Yq">Resolution of singularities</a> in characteristic p {\displaystyle p} </li></ul> <h4>Covering and packing</h4> <ul><li><a href="/facts/Borsuk%27s_conjecture/gD4tHhsU">Borsuk's problem</a> on upper and lower bounds for the number of smaller-diameter subsets needed to cover a <a href="/facts/Bounded_set/yCldjtoy">bounded</a> <i>n</i>-dimensional set.</li> <li>The <a href="/facts/Covering_problem_of_Rado/8RKCRcta">covering problem of Rado</a>: 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?<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a></li> <li>The <a href="/facts/Circle_packing_in_an_equilateral_triangle/q0PlUTdB">Erdős–Oler conjecture</a>: when n {\displaystyle n} is a <a href="/facts/Triangular_number/KHonHtQJ">triangular number</a>, packing n − 1 {\displaystyle n-1} circles in an equilateral triangle requires a triangle of the same size as packing n {\displaystyle n} circles.<a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a></li> <li>The <a href="/facts/Disk_covering_problem/LDHTedBG">disk covering problem</a> abount finding the smallest <a href="/facts/Real_number/R02gw5Pb">real number</a> r ( n ) {\displaystyle r(n)} such that n {\displaystyle n} <a href="/facts/Disk_(mathematics)/DgMVr97H">disks</a> of radius r ( n ) {\displaystyle r(n)} can be arranged in such a way as to cover the <a href="/facts/Unit_disk/phrUu7yC">unit disk</a>.</li> <li>The <a href="/facts/Kissing_number_problem/UE3mtV0d">kissing number problem</a> for dimensions other than 1, 2, 3, 4, 8 and 24<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a></li> <li><a href="/facts/Reinhardt%27s_conjecture/K7sfhVRU">Reinhardt's conjecture</a>: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a></li> <li><a href="/facts/Sphere_packing/NY2FKyMf">Sphere packing</a> 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.</li> <li><a href="/facts/Square_packing_in_a_square/xa0DnStP">Square packing in a square</a>: what is the asymptotic growth rate of wasted space?<a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a></li> <li><a href="/facts/Ulam%27s_packing_conjecture/TMYvNCyQ">Ulam's packing conjecture</a> about the identity of the worst-packing convex solid<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a></li> <li>The <a href="/facts/Tammes_problem/9LR5FVWe">Tammes problem</a> for numbers of nodes greater than 14 (except 24).<a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a></li></ul> <h4>Differential geometry</h4> <p class="note">Main article: <a href="/facts/Differential_geometry/c04XmyLu">Differential geometry</a></p> <ul><li>The <a href="/facts/Spherical_Bernstein%27s_problem/sAbkGbjj">spherical Bernstein's problem</a>, a generalization of <a href="/facts/Bernstein%27s_problem/FPBA1fcp">Bernstein's problem</a></li> <li><a href="/facts/Carath%C3%A9odory_conjecture/vvHqFzE7">Carathéodory conjecture</a>: any convex, closed, and twice-differentiable surface in three-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> admits at least two <a href="/facts/Umbilical_point/kGaaHg4f">umbilical points</a>.</li> <li><a href="/facts/Cartan%E2%80%93Hadamard_conjecture/lkaDnDbT">Cartan–Hadamard conjecture</a>: can the classical <a href="/facts/Isoperimetric_inequality/u02PG11C">isoperimetric inequality</a> for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as <a href="/facts/Hadamard_manifold/3EqgJFTG">Cartan–Hadamard manifolds</a>?</li> <li><a href="/facts/Chern%27s_conjecture_(affine_geometry)/nYejD9W3">Chern's conjecture (affine geometry)</a> that the <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> of a <a href="/facts/Closed_manifold/jMG6G6Bl">compact</a> <a href="/facts/Affine_manifold/6wVyaEhT">affine manifold</a> vanishes.</li> <li><a href="/facts/Chern%27s_conjecture_for_hypersurfaces_in_spheres/ADRGl9WV">Chern's conjecture for hypersurfaces in spheres</a>, a number of closely related conjectures.</li> <li>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.<a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a></li> <li>The <a href="/facts/Filling_area_conjecture/wJ5S5oVg">filling area conjecture</a>, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length<a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a></li> <li>The <a href="/facts/Hopf_conjecture/bMjgk3Gu">Hopf conjectures</a> relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds<a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a></li> <li>Osserman conjecture: that every Osserman manifold is either <a href="/facts/Flat_manifold/YBQfE8mn">flat</a> or locally <a href="/facts/Isometry/K5wX8ah6">isometric</a> to a rank-one <a href="/facts/Symmetric_space/AWYuHfNp">symmetric space</a><a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a></li> <li><a href="/facts/Yau%27s_conjecture_on_the_first_eigenvalue/5SEWWaGE">Yau's conjecture on the first eigenvalue</a> that the first <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalue</a> for the <a href="/facts/Laplace%E2%80%93Beltrami_operator/AJmyUpWz">Laplace–Beltrami operator</a> on an embedded <a href="/facts/Minimal_surface/Fy7VkrLQ">minimal hypersurface</a> of S n + 1 {\displaystyle S^{n+1}} is n {\displaystyle n} .</li></ul> <h4>Discrete geometry</h4> <p class="note">Main article: <a href="/facts/Discrete_geometry/Ixnl3GBc">Discrete geometry</a></p> <ul><li>The big-line-big-clique conjecture on the existence of either many collinear points or many mutually visible points in large planar point sets<a class="footnote-ref" id="fnref:59" href="#fn:59"><sup>59</sup></a></li> <li>The <a href="/facts/Hadwiger_conjecture_(combinatorial_geometry)/wTFeLzXn">Hadwiger conjecture</a> on covering <i>n</i>-dimensional convex bodies with at most 2<i>n</i> smaller copies<a class="footnote-ref" id="fnref:60" href="#fn:60"><sup>60</sup></a></li> <li>Solving the <a href="/facts/Happy_ending_problem/qVHgkcDg">happy ending problem</a> for arbitrary n {\displaystyle n} <a class="footnote-ref" id="fnref:61" href="#fn:61"><sup>61</sup></a></li> <li>Improving lower and upper bounds for the <a href="/facts/Heilbronn_triangle_problem/tC5soztj">Heilbronn triangle problem</a>.</li> <li><a href="/facts/Kalai%27s_3%5Ed_conjecture/BsrrsrpE">Kalai's 3<i>d</i> conjecture</a> on the least possible number of faces of <a href="/facts/Point_symmetry/UzswAm8N">centrally symmetric</a> <a href="/facts/Polytopes/a76nXrL0">polytopes</a>.<a class="footnote-ref" id="fnref:62" href="#fn:62"><sup>62</sup></a></li> <li>The <a href="/facts/Kobon_triangle_problem/9Wjob53q">Kobon triangle problem</a> on triangles in line arrangements<a class="footnote-ref" id="fnref:63" href="#fn:63"><sup>63</sup></a></li> <li>The <a href="/facts/Kusner_conjecture/Tp8tCLBJ">Kusner conjecture</a>: at most 2 d {\displaystyle 2d} points can be equidistant in L 1 {\displaystyle L^{1}} spaces<a class="footnote-ref" id="fnref:64" href="#fn:64"><sup>64</sup></a></li> <li>The <a href="/facts/McMullen_problem/bpsRXYnD">McMullen problem</a> on projectively transforming sets of points into <a href="/facts/Convex_position/lzW3leK9">convex position</a><a class="footnote-ref" id="fnref:65" href="#fn:65"><sup>65</sup></a></li> <li><a href="/facts/Opaque_forest_problem/dhl9Q8bY">Opaque forest problem</a> on finding <a href="/facts/Opaque_set/dhl9Q8bY">opaque sets</a> for various planar shapes</li> <li><a href="/facts/Unit_distance_graph/8y4Ctq53">How many unit distances</a> can be determined by a set of n points in the Euclidean plane?<a class="footnote-ref" id="fnref:66" href="#fn:66"><sup>66</sup></a></li> <li>Finding matching upper and lower bounds for <a href="/facts/K-set_(geometry)/kp7Vobgu"><i>k</i>-sets</a> and halving lines<a class="footnote-ref" id="fnref:67" href="#fn:67"><sup>67</sup></a></li> <li><a href="/facts/Tripod_packing/utTc84F5">Tripod packing</a>:<a class="footnote-ref" id="fnref:68" href="#fn:68"><sup>68</sup></a> how many tripods can have their apexes packed into a given cube?</li></ul> <h4>Euclidean geometry</h4> <p class="note">Main article: <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a></p> <ul><li>The <a href="/facts/Atiyah_conjecture_on_configurations/nfWlA9pN">Atiyah conjecture on configurations</a> on the invertibility of a certain n {\displaystyle n} -by- n {\displaystyle n} matrix depending on n {\displaystyle n} points in R 3 {\displaystyle \mathbb {R} ^{3}} <a class="footnote-ref" id="fnref:69" href="#fn:69"><sup>69</sup></a></li> <li><a href="/facts/Bellman%27s_lost-in-a-forest_problem/db1SMC12">Bellman's lost-in-a-forest problem</a> – 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<a class="footnote-ref" id="fnref:70" href="#fn:70"><sup>70</sup></a></li> <li><a href="/facts/Borromean_rings/C16wTE0j">Borromean rings</a> — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?<a class="footnote-ref" id="fnref:71" href="#fn:71"><sup>71</sup></a></li> <li><a href="/facts/Connelly%E2%80%99s_blooming_conjecture/kb1HpMSH">Connelly’s blooming conjecture</a>: Does every net of a convex polyhedron have a <a href="/facts/Blooming_(geometry)/kb1HpMSH">blooming</a>?<a class="footnote-ref" id="fnref:72" href="#fn:72"><sup>72</sup></a></li> <li>Danzer's problem and Conway's dead fly problem – do <a href="/facts/Danzer_set/86oEfApl">Danzer sets</a> of bounded density or bounded separation exist?<a class="footnote-ref" id="fnref:73" href="#fn:73"><sup>73</sup></a></li> <li><a href="/facts/Dissection_into_orthoschemes/JFF1SktF">Dissection into orthoschemes</a> – is it possible for <a href="/facts/Simplex/0FCfNRN8">simplices</a> of every dimension?<a class="footnote-ref" id="fnref:74" href="#fn:74"><sup>74</sup></a></li> <li><a href="/facts/Ehrhart%27s_volume_conjecture/4L4V431f">Ehrhart's volume conjecture</a>: a convex body K {\displaystyle K} in n {\displaystyle n} dimensions containing a single lattice point in its interior as its <a href="/facts/Center_of_mass/n4Lr9GAz">center of mass</a> cannot have volume greater than ( n + 1 ) n / n ! {\displaystyle (n+1)^{n}/n!} </li> <li><a href="/facts/Falconer%27s_conjecture/AlG6cb8c">Falconer's conjecture</a>: sets of Hausdorff dimension greater than d / 2 {\displaystyle d/2} in R d {\displaystyle \mathbb {R} ^{d}} must have a distance set of nonzero <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a><a class="footnote-ref" id="fnref:75" href="#fn:75"><sup>75</sup></a></li> <li>The values of the <a href="/facts/Hermite_constant/nq713k79">Hermite constants</a> for dimensions other than 1–8 and 24</li> <li>What is the lowest number of faces possible for a <a href="/facts/Holyhedron/aGyvR6Ax">holyhedron</a>?</li> <li><a href="/facts/Inscribed_square_problem/Mbrb9cwN">Inscribed square problem</a>, also known as <a href="/facts/Toeplitz%27_conjecture/Mbrb9cwN">Toeplitz' conjecture</a> and the square peg problem – does every <a href="/facts/Jordan_curve/zKpty4QT">Jordan curve</a> have an inscribed square?<a class="footnote-ref" id="fnref:76" href="#fn:76"><sup>76</sup></a></li> <li>The <a href="/facts/Kakeya_conjecture/an2MrLeQ">Kakeya conjecture</a> – do n {\displaystyle n} -dimensional sets that contain a unit line segment in every direction necessarily have <a href="/facts/Hausdorff_dimension/rtX9ryrp">Hausdorff dimension</a> and <a href="/facts/Minkowski_dimension/lDLdIezz">Minkowski dimension</a> equal to n {\displaystyle n} ?<a class="footnote-ref" id="fnref:77" href="#fn:77"><sup>77</sup></a></li> <li>The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the <a href="/facts/Weaire%E2%80%93Phelan_structure/QoXHtJDk">Weaire–Phelan structure</a> as a solution to the Kelvin problem<a class="footnote-ref" id="fnref:78" href="#fn:78"><sup>78</sup></a></li> <li><a href="/facts/Lebesgue%27s_universal_covering_problem/p8gnf8Qm">Lebesgue's universal covering problem</a> on the minimum-area convex shape in the plane that can cover any shape of diameter one<a class="footnote-ref" id="fnref:79" href="#fn:79"><sup>79</sup></a></li> <li><a href="/facts/Mahler_volume/Ru1mydQy">Mahler's conjecture</a> on the product of the volumes of a <a href="/facts/Central_symmetry/UzswAm8N">centrally symmetric</a> <a href="/facts/Convex_body/90knDx0u">convex body</a> and its <a href="/facts/Polar_set/JY45pfbt">polar</a>.<a class="footnote-ref" id="fnref:80" href="#fn:80"><sup>80</sup></a></li> <li><a href="/facts/Moser%27s_worm_problem/nq47sn6j">Moser's worm problem</a> – what is the smallest area of a shape that can cover every unit-length curve in the plane?<a class="footnote-ref" id="fnref:81" href="#fn:81"><sup>81</sup></a></li> <li>The <a href="/facts/Moving_sofa_problem/4e2wlFvI">moving sofa problem</a> – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?<a class="footnote-ref" id="fnref:82" href="#fn:82"><sup>82</sup></a></li> <li>Does every convex polyhedron have <a href="/facts/Prince_Rupert%27s_cube/9kftj9rT">Rupert's property</a>?<a class="footnote-ref" id="fnref:83" href="#fn:83"><sup>83</sup></a><a class="footnote-ref" id="fnref:84" href="#fn:84"><sup>84</sup></a></li> <li><a href="/facts/Shephard%27s_conjecture/lU8fmGJs">Shephard's problem (a.k.a. Dürer's conjecture)</a> – does every <a href="/facts/Convex_polyhedron/2pRHcRgC">convex polyhedron</a> have a <a href="/facts/Net_(polyhedron)/lU8fmGJs">net</a>, or simple edge-unfolding?<a class="footnote-ref" id="fnref:85" href="#fn:85"><sup>85</sup></a><a class="footnote-ref" id="fnref:86" href="#fn:86"><sup>86</sup></a></li> <li>Is there a non-convex polyhedron without self-intersections with <a href="/facts/Szilassi_polyhedron/NFcNem6V">more than seven faces</a>, all of which share an edge with each other?</li> <li>The <a href="/facts/Thomson_problem/eyDlFwy4">Thomson problem</a> – what is the minimum energy configuration of n {\displaystyle n} mutually-repelling particles on a unit sphere?<a class="footnote-ref" id="fnref:87" href="#fn:87"><sup>87</sup></a></li> <li>Convex <a href="/facts/Uniform_5-polytope/mqxsoqwY">uniform 5-polytopes</a> – find and classify the complete set of these shapes<a class="footnote-ref" id="fnref:88" href="#fn:88"><sup>88</sup></a></li></ul> <h3>Graph theory</h3> <p class="note">Main article: <a href="/facts/Graph_theory/VVPCd4TX">Graph theory</a></p> <h4>Algebraic graph theory</h4> <ul><li><a href="/facts/Babai%27s_problem/BkSwHgld">Babai's problem</a>: which groups are Babai invariant groups?</li> <li><a href="/facts/Brouwer%27s_conjecture/JTfLq3ML">Brouwer's conjecture</a> on upper bounds for sums of <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of <a href="/facts/Laplacian_matrix/Iwea2mm9">Laplacians</a> of graphs in terms of their number of edges</li></ul> <h4>Games on graphs</h4> <ul><li>Does there exist a graph G {\displaystyle G} such that the <a href="/facts/Dominating_set/3xX4kSnP">dominating number</a> γ ( G ) {\displaystyle \gamma (G)} equals the <a href="/facts/Eternal_dominating_set/Q2Pdkfv9">eternal dominating number</a> γ {\displaystyle \gamma } ∞ ( G ) {\displaystyle (G)} of G {\displaystyle G} and γ ( G ) {\displaystyle \gamma (G)} is less than the <a href="/facts/Clique_cover/tXT9GSVq">clique covering number</a> of G {\displaystyle G} ? <a class="footnote-ref" id="fnref:89" href="#fn:89"><sup>89</sup></a></li> <li><a href="/facts/Graham%27s_pebbling_conjecture/B1aM1jlU">Graham's pebbling conjecture</a> on the pebbling number of Cartesian products of graphs<a class="footnote-ref" id="fnref:90" href="#fn:90"><sup>90</sup></a></li> <li>Meyniel's conjecture that <a href="/facts/Cop_number/BvSc2QK3">cop number</a> is O ( n ) {\displaystyle O({\sqrt {n}})} <a class="footnote-ref" id="fnref:91" href="#fn:91"><sup>91</sup></a></li> <li>Suppose Alice has a winning strategy for the <a href="/facts/Graph_coloring_game/m7oue8TC">vertex coloring game</a> on a graph G {\displaystyle G} with k {\displaystyle k} colors. Does she have one for k + 1 {\displaystyle k+1} colors?<a class="footnote-ref" id="fnref:92" href="#fn:92"><sup>92</sup></a></li></ul> <h4>Graph coloring and labeling</h4> <ul><li>The <a href="/facts/Graph_factorization/2hx3n7c3">1-factorization conjecture</a> that if n {\displaystyle n} is odd or even and k ≥ n , n − 1 {\displaystyle k\geq n,n-1} respectively, then a k {\displaystyle k} -<a href="/facts/Regular_graph/j9dSoLE6">regular graph</a> with 2 n {\displaystyle 2n} vertices is <a href="/facts/Graph_factorization/2hx3n7c3">1-factorable</a>. <ul><li>The <a href="/facts/Graph_factorization/2hx3n7c3">perfect 1-factorization conjecture</a> that every <a href="/facts/Complete_graph/AYD6hUqI">complete graph</a> on an even number of vertices admits a <a href="/facts/Graph_factorization/2hx3n7c3">perfect 1-factorization</a>.</li></ul></li> <li><a href="/facts/Cereceda%27s_conjecture/fVuNlLaI">Cereceda's conjecture</a> on the diameter of the space of colorings of degenerate graphs<a class="footnote-ref" id="fnref:93" href="#fn:93"><sup>93</sup></a></li> <li>The <a href="/facts/Earth%E2%80%93Moon_problem/enLuc2NJ">Earth–Moon problem</a>: what is the maximum chromatic number of biplanar graphs?<a class="footnote-ref" id="fnref:94" href="#fn:94"><sup>94</sup></a></li> <li>The <a href="/facts/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture/6usGHXjs">Erdős–Faber–Lovász conjecture</a> on coloring unions of cliques<a class="footnote-ref" id="fnref:95" href="#fn:95"><sup>95</sup></a></li> <li>The <a href="/facts/Graceful_labeling/Do7bxMnc">graceful tree conjecture</a> that every tree admits a graceful labeling <ul><li><a href="/facts/Graceful_labeling/Do7bxMnc">Rosa's conjecture</a> that all <a href="/facts/Cactus_graph/s85ekIxz">triangular cacti</a> are graceful or nearly-graceful</li></ul></li> <li>The <a href="/facts/Gy%C3%A1rf%C3%A1s%E2%80%93Sumner_conjecture/uzx8TzSb">Gyárfás–Sumner conjecture</a> on χ-boundedness of graphs with a forbidden induced tree<a class="footnote-ref" id="fnref:96" href="#fn:96"><sup>96</sup></a></li> <li>The <a href="/facts/Hadwiger_conjecture_(graph_theory)/9m1YSAKy">Hadwiger conjecture</a> relating coloring to clique minors<a class="footnote-ref" id="fnref:97" href="#fn:97"><sup>97</sup></a></li> <li>The <a href="/facts/Hadwiger%E2%80%93Nelson_problem/zxjGbKhl">Hadwiger–Nelson problem</a> on the chromatic number of unit distance graphs<a class="footnote-ref" id="fnref:98" href="#fn:98"><sup>98</sup></a></li> <li><a href="/facts/Petersen_graph/lLRxXpQs">Jaeger's Petersen-coloring conjecture</a>: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph<a class="footnote-ref" id="fnref:99" href="#fn:99"><sup>99</sup></a></li> <li>The <a href="/facts/List_coloring_conjecture/FKQKVMUA">list coloring conjecture</a>: for every graph, the list chromatic index equals the chromatic index<a class="footnote-ref" id="fnref:100" href="#fn:100"><sup>100</sup></a></li> <li>The <a href="/facts/Overfull_graph/QUeaecdE">overfull conjecture</a> that a graph with maximum degree Δ ( G ) ≥ n / 3 {\displaystyle \Delta (G)\geq n/3} is <a href="/facts/Vizing%27s_theorem/hHtEtK9t">class 2</a> if and only if it has an <a href="/facts/Overfull_graph/QUeaecdE">overfull subgraph</a> S {\displaystyle S} satisfying Δ ( S ) = Δ ( G ) {\displaystyle \Delta (S)=\Delta (G)} .</li> <li>The <a href="/facts/Total_coloring_conjecture/CwxcyopP">total coloring conjecture</a> of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree<a class="footnote-ref" id="fnref:101" href="#fn:101"><sup>101</sup></a></li></ul> <h4>Graph drawing and embedding</h4> <ul><li>The <a href="/facts/Albertson_conjecture/gN8afN37">Albertson conjecture</a>: the crossing number can be lower-bounded by the crossing number of a <a href="/facts/Complete_graph/AYD6hUqI">complete graph</a> with the same <a href="/facts/Chromatic_number/J6HoBfP0">chromatic number</a><a class="footnote-ref" id="fnref:102" href="#fn:102"><sup>102</sup></a></li> <li><a href="/facts/Conway%27s_thrackle_conjecture/KAGs310p">Conway's thrackle conjecture</a><a class="footnote-ref" id="fnref:103" href="#fn:103"><sup>103</sup></a> that <a href="/facts/Thrackle/KAGs310p">thrackles</a> cannot have more edges than vertices</li> <li>The <a href="/facts/GNRS_conjecture/7v3Yjf32">GNRS conjecture</a> on whether minor-closed graph families have ℓ 1 {\displaystyle \ell _{1}} embeddings with bounded distortion<a class="footnote-ref" id="fnref:104" href="#fn:104"><sup>104</sup></a></li> <li><a href="/facts/Harborth%27s_conjecture/F5WRSGSf">Harborth's conjecture</a>: every planar graph can be drawn with integer edge lengths<a class="footnote-ref" id="fnref:105" href="#fn:105"><sup>105</sup></a></li> <li><a href="/facts/Negami%27s_conjecture/TX1NupDf">Negami's conjecture</a> on projective-plane embeddings of graphs with planar covers<a class="footnote-ref" id="fnref:106" href="#fn:106"><sup>106</sup></a></li> <li>The <a href="/facts/Greedy_embedding/3s6jdNPM">strong Papadimitriou–Ratajczak conjecture</a>: every polyhedral graph has a convex greedy embedding<a class="footnote-ref" id="fnref:107" href="#fn:107"><sup>107</sup></a></li> <li><a href="/facts/Tur%C3%A1n%27s_brick_factory_problem/o2770HsJ">Turán's brick factory problem</a> – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?<a class="footnote-ref" id="fnref:108" href="#fn:108"><sup>108</sup></a></li></ul> <ul><li><a href="/facts/Universal_point_set/dQ0mmdme">Universal point sets</a> of subquadratic size for planar graphs<a class="footnote-ref" id="fnref:109" href="#fn:109"><sup>109</sup></a></li></ul> <h4>Restriction of graph parameters</h4> <ul><li><a href="/facts/Conway%27s_99-graph_problem/HlumDAzs">Conway's 99-graph problem</a>: does there exist a <a href="/facts/Strongly_regular_graph/0GbdLwSd">strongly regular graph</a> with parameters (99,14,1,2)?<a class="footnote-ref" id="fnref:110" href="#fn:110"><sup>110</sup></a></li> <li><a href="/facts/Degree_diameter_problem/uaGL4qFr">Degree diameter problem</a>: given two positive integers d , k {\displaystyle d,k} , what is the largest graph of diameter k {\displaystyle k} such that all vertices have degrees at most d {\displaystyle d} ?</li> <li>Jørgensen's conjecture that every 6-vertex-connected <i>K</i>6-minor-free graph is an <a href="/facts/Apex_graph/UzXxlwph">apex graph</a><a class="footnote-ref" id="fnref:111" href="#fn:111"><sup>111</sup></a></li> <li>Does a <a href="/facts/Moore_graph/obJmADTb">Moore graph</a> with girth 5 and degree 57 exist?<a class="footnote-ref" id="fnref:112" href="#fn:112"><sup>112</sup></a></li> <li>Do there exist infinitely many <a href="/facts/Strongly_regular_graph/0GbdLwSd">strongly regular</a> <a href="/facts/Geodetic_graph/JQoNejMN">geodetic graphs</a>, or any strongly regular geodetic graphs that are not Moore graphs?<a class="footnote-ref" id="fnref:113" href="#fn:113"><sup>113</sup></a></li></ul> <h4>Subgraphs</h4> <ul><li><a href="/facts/Barnette%27s_conjecture/6dGR5Ebx">Barnette's conjecture</a>: every cubic bipartite three-connected planar graph has a Hamiltonian cycle<a class="footnote-ref" id="fnref:114" href="#fn:114"><sup>114</sup></a></li> <li><a href="/facts/Gilbert%E2%80%93Pollack_conjecture_on_the_Steiner_ratio_of_the_Euclidean_plane/mF4YjpIP">Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane</a> that the Steiner ratio is 3 / 2 {\displaystyle {\sqrt {3}}/2} </li> <li><a href="/facts/Graph_toughness/92anGVk5">Chvátal's toughness conjecture</a>, that there is a number t such that every t-tough graph is Hamiltonian<a class="footnote-ref" id="fnref:115" href="#fn:115"><sup>115</sup></a></li> <li>The <a href="/facts/Cycle_double_cover_conjecture/vtGSVkxz">cycle double cover conjecture</a>: every bridgeless graph has a family of cycles that includes each edge twice<a class="footnote-ref" id="fnref:116" href="#fn:116"><sup>116</sup></a></li> <li>The <a href="/facts/Erd%C5%91s%E2%80%93Gy%C3%A1rf%C3%A1s_conjecture/JPX36lmC">Erdős–Gyárfás conjecture</a> on cycles with power-of-two lengths in cubic graphs<a class="footnote-ref" id="fnref:117" href="#fn:117"><sup>117</sup></a></li> <li>The <a href="/facts/Erd%C5%91s%E2%80%93Hajnal_conjecture/6u6084rC">Erdős–Hajnal conjecture</a> on large cliques or independent sets in graphs with a forbidden induced subgraph<a class="footnote-ref" id="fnref:118" href="#fn:118"><sup>118</sup></a></li> <li>The <a href="/facts/Linear_arboricity/KfCGmdDG">linear arboricity</a> conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree<a class="footnote-ref" id="fnref:119" href="#fn:119"><sup>119</sup></a></li> <li>The <a href="/facts/Lov%C3%A1sz_conjecture/9oznzvAB">Lovász conjecture</a> on Hamiltonian paths in symmetric graphs<a class="footnote-ref" id="fnref:120" href="#fn:120"><sup>120</sup></a></li> <li>The <a href="/facts/Oberwolfach_problem/DhC1IE0m">Oberwolfach problem</a> 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.<a class="footnote-ref" id="fnref:121" href="#fn:121"><sup>121</sup></a></li> <li>What is the largest possible <a href="/facts/Pathwidth/Gf5S2byH">pathwidth</a> of an n-vertex <a href="/facts/Cubic_graph/6M32WUIJ">cubic graph</a>?<a class="footnote-ref" id="fnref:122" href="#fn:122"><sup>122</sup></a></li> <li>The <a href="/facts/Reconstruction_conjecture/sYv3VVEI">reconstruction conjecture</a> and <a href="/facts/New_digraph_reconstruction_conjecture/JYMVMmUA">new digraph reconstruction conjecture</a> on whether a graph is uniquely determined by its vertex-deleted subgraphs.<a class="footnote-ref" id="fnref:123" href="#fn:123"><sup>123</sup></a><a class="footnote-ref" id="fnref:124" href="#fn:124"><sup>124</sup></a></li> <li>The <a href="/facts/Snake-in-the-box/g72CeAYK">snake-in-the-box</a> problem: what is the longest possible <a href="/facts/Induced_path/zfcIx9Sw">induced path</a> in an n {\displaystyle n} -dimensional <a href="/facts/Hypercube/q2mRPbIz">hypercube</a> graph?</li> <li><a href="/facts/Sumner%27s_conjecture/LjAquw0x">Sumner's conjecture</a>: does every ( 2 n − 2 ) {\displaystyle (2n-2)} -vertex tournament contain as a subgraph every n {\displaystyle n} -vertex oriented tree?<a class="footnote-ref" id="fnref:125" href="#fn:125"><sup>125</sup></a></li> <li><a href="/facts/Szymanski%27s_conjecture/u3zV0GRD">Szymanski's conjecture</a>: every <a href="/facts/Permutation/JSw9ukmv">permutation</a> on the n {\displaystyle n} -dimensional doubly-<a href="/facts/Directed_graph/YBdfQ0um">directed</a> <a href="/facts/Hypercube_graph/pwfGEl9J">hypercube graph</a> can be routed with edge-disjoint <a href="/facts/Path_(graph_theory)/WMsq21Nf">paths</a>.</li> <li>Tuza's conjecture: if the maximum number of disjoint triangles is ν {\displaystyle \nu } , can all triangles be hit by a set of at most 2 ν {\displaystyle 2\nu } edges?<a class="footnote-ref" id="fnref:126" href="#fn:126"><sup>126</sup></a></li> <li><a href="/facts/Vizing%27s_conjecture/2wIoCVCw">Vizing's conjecture</a> on the <a href="/facts/Domination_number/3xX4kSnP">domination number</a> of <a href="/facts/Cartesian_product_of_graphs/wC4JJdyU">cartesian products of graphs</a><a class="footnote-ref" id="fnref:127" href="#fn:127"><sup>127</sup></a></li> <li><a href="/facts/Zarankiewicz_problem/388M38rX">Zarankiewicz problem</a>: how many edges can there be in a <a href="/facts/Bipartite_graph/xWcXV9MB">bipartite graph</a> on a given number of vertices with no <a href="/facts/Complete_bipartite_graph/C1RYIdpX">complete bipartite subgraphs</a> of a given size?</li></ul> <h4>Word-representation of graphs</h4> <ul><li>Are there any graphs on <i>n</i> vertices whose <a href="/facts/Word-representable_graph/6bIHhngq">representation</a> requires more than floor(<i>n</i>/2) copies of each letter?<a class="footnote-ref" id="fnref:128" href="#fn:128"><sup>128</sup></a><a class="footnote-ref" id="fnref:129" href="#fn:129"><sup>129</sup></a><a class="footnote-ref" id="fnref:130" href="#fn:130"><sup>130</sup></a><a class="footnote-ref" id="fnref:131" href="#fn:131"><sup>131</sup></a></li> <li>Characterise (non-)<a href="/facts/Word-representable_graph/6bIHhngq">word-representable</a> <a href="/facts/Planar_graph/plREagr9">planar graphs</a><a class="footnote-ref" id="fnref:132" href="#fn:132"><sup>132</sup></a><a class="footnote-ref" id="fnref:133" href="#fn:133"><sup>133</sup></a><a class="footnote-ref" id="fnref:134" href="#fn:134"><sup>134</sup></a><a class="footnote-ref" id="fnref:135" href="#fn:135"><sup>135</sup></a></li> <li>Characterise <a href="/facts/Word-representable_graph/6bIHhngq">word-representable graphs</a> in terms of (induced) forbidden subgraphs.<a class="footnote-ref" id="fnref:136" href="#fn:136"><sup>136</sup></a><a class="footnote-ref" id="fnref:137" href="#fn:137"><sup>137</sup></a><a class="footnote-ref" id="fnref:138" href="#fn:138"><sup>138</sup></a><a class="footnote-ref" id="fnref:139" href="#fn:139"><sup>139</sup></a></li> <li>Characterise <a href="/facts/Word-representable_graph/6bIHhngq">word-representable</a> near-triangulations containing the complete graph <i>K</i>4 (such a characterisation is known for <i>K</i>4-free planar graphs<a class="footnote-ref" id="fnref:140" href="#fn:140"><sup>140</sup></a>)</li> <li>Classify graphs with representation number 3, that is, graphs that can be <a href="/facts/Word-representable_graph/6bIHhngq">represented</a> using 3 copies of each letter, but cannot be represented using 2 copies of each letter<a class="footnote-ref" id="fnref:141" href="#fn:141"><sup>141</sup></a></li> <li>Is it true that out of all <a href="/facts/Bipartite_graph/xWcXV9MB">bipartite graphs</a>, <a href="/facts/Crown_graph/aauWDwaX">crown graphs</a> require longest word-representants?<a class="footnote-ref" id="fnref:142" href="#fn:142"><sup>142</sup></a></li> <li>Is the <a href="/facts/Line_graph/RWNxSy5j">line graph</a> of a non-<a href="/facts/Word-representable_graph/6bIHhngq">word-representable</a> graph always non-<a href="/facts/Word-representable_graph/6bIHhngq">word-representable</a>?<a class="footnote-ref" id="fnref:143" href="#fn:143"><sup>143</sup></a><a class="footnote-ref" id="fnref:144" href="#fn:144"><sup>144</sup></a><a class="footnote-ref" id="fnref:145" href="#fn:145"><sup>145</sup></a><a class="footnote-ref" id="fnref:146" href="#fn:146"><sup>146</sup></a></li> <li>Which (hard) problems on graphs can be translated to words <a href="/facts/Word-representable_graph/6bIHhngq">representing</a> them and solved on words (efficiently)?<a class="footnote-ref" id="fnref:147" href="#fn:147"><sup>147</sup></a><a class="footnote-ref" id="fnref:148" href="#fn:148"><sup>148</sup></a><a class="footnote-ref" id="fnref:149" href="#fn:149"><sup>149</sup></a><a class="footnote-ref" id="fnref:150" href="#fn:150"><sup>150</sup></a></li></ul> <h4>Miscellaneous graph theory</h4> <ul><li>The <a href="/facts/Implicit_graph_conjecture/ERbgYekg">implicit graph conjecture</a> on the existence of implicit representations for slowly-growing <a href="/facts/Hereditary_property/J64xYUGG">hereditary families of graphs</a><a class="footnote-ref" id="fnref:151" href="#fn:151"><sup>151</sup></a></li> <li><a href="/facts/Ryser%27s_conjecture/UY9gMxqo">Ryser's conjecture</a> relating the maximum <a href="/facts/Matching_in_hypergraphs/3tt40LSh">matching</a> size and minimum <a href="/facts/Vertex_cover_in_hypergraphs/9HW3kQQe">transversal</a> size in <a href="/facts/Hypergraph/qIR2o2I3">hypergraphs</a></li> <li>The <a href="/facts/Second_neighborhood_problem/AwB5a557">second neighborhood problem</a>: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?<a class="footnote-ref" id="fnref:152" href="#fn:152"><sup>152</sup></a></li> <li><a href="/facts/Sidorenko%27s_conjecture/jxzA8ffr">Sidorenko's conjecture</a> on <a href="/facts/Homomorphism_density/V5cHzcK5">homomorphism densities</a> of graphs in <a href="/facts/Graphon/ezqevlSv">graphons</a></li> <li>Tutte's conjectures: <ul><li>every bridgeless graph has a <a href="/facts/Nowhere-zero_flows/NYW9vQhk">nowhere-zero 5-flow</a><a class="footnote-ref" id="fnref:153" href="#fn:153"><sup>153</sup></a></li> <li>every <a href="/facts/Petersen_graph/lLRxXpQs">Petersen</a>-<a href="/facts/Graph_minor/qn5eI22T">minor</a>-free bridgeless graph has a nowhere-zero 4-flow<a class="footnote-ref" id="fnref:154" href="#fn:154"><sup>154</sup></a></li></ul></li> <li><a href="/facts/Woodall%27s_conjecture/ItK9DdBx">Woodall's conjecture</a> that the minimum number of edges in a <a href="/facts/Dicut/SUvTfbbl">dicut</a> of a <a href="/facts/Directed_graph/YBdfQ0um">directed graph</a> is equal to the maximum number of disjoint <a href="/facts/Dijoin/Blx6tyh4">dijoins</a></li></ul> <h3>Model theory and formal languages</h3> <p class="note">Main articles: <a href="/facts/Model_theory/vLWVV8sQ">Model theory</a> and <a href="/facts/Formal_languages/crDTyP8q">formal languages</a></p> <ul><li>The <a href="/facts/Stable_group/DIoEvjWl">Cherlin–Zilber conjecture</a>: A simple group whose first-order theory is <a href="/facts/Stable_theory/0bVGhbeK">stable</a> in ℵ 0 {\displaystyle \aleph _{0}} is a simple algebraic group over an algebraically closed field.</li> <li><a href="/facts/Generalized_star_height_problem/GL7RWU3N">Generalized star height problem</a>: can all <a href="/facts/Regular_language/ahxw67T1">regular languages</a> be expressed using <a href="/facts/Regular_expression/KFL3veHX">generalized regular expressions</a> with limited nesting depths of <a href="/facts/Kleene_star/I6SX0f2Y">Kleene stars</a>?</li> <li>For which number fields does <a href="/facts/Hilbert%27s_tenth_problem/EaGPklEc">Hilbert's tenth problem</a> hold?</li> <li>Kueker's conjecture<a class="footnote-ref" id="fnref:155" href="#fn:155"><sup>155</sup></a></li> <li>The main gap conjecture, e.g. for uncountable <a href="/facts/First_order_theory/HAse4SIg">first order theories</a>, for <a href="/facts/Abstract_elementary_class/kI0Ea3DJ">AECs</a>, and for ℵ 1 {\displaystyle \aleph _{1}} -saturated models of a countable theory.<a class="footnote-ref" id="fnref:156" href="#fn:156"><sup>156</sup></a></li> <li>Shelah's categoricity conjecture for L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.<a class="footnote-ref" id="fnref:157" href="#fn:157"><sup>157</sup></a></li> <li>Shelah's eventual categoricity conjecture: For every cardinal λ {\displaystyle \lambda } there exists a cardinal μ ( λ ) {\displaystyle \mu (\lambda )} such that if an <a href="/facts/Abstract_elementary_class/kI0Ea3DJ">AEC</a> <i>K</i> with LS(<i>K</i>) ≤ λ {\displaystyle {}\leq \lambda } is categorical in a cardinal above μ ( λ ) {\displaystyle \mu (\lambda )} then it is categorical in all cardinals above μ ( λ ) {\displaystyle \mu (\lambda )} .<a class="footnote-ref" id="fnref:158" href="#fn:158"><sup>158</sup></a><a class="footnote-ref" id="fnref:159" href="#fn:159"><sup>159</sup></a></li> <li>The stable field conjecture: every infinite field with a <a href="/facts/Stable_theory/0bVGhbeK">stable</a> first-order theory is separably closed.</li> <li>The stable forking conjecture for simple theories<a class="footnote-ref" id="fnref:160" href="#fn:160"><sup>160</sup></a></li> <li><a href="/facts/Tarski%27s_exponential_function_problem/2uHGJHKU">Tarski's exponential function problem</a>: is the <a href="/facts/Theory_(mathematical_logic)/mYe5gQpI">theory</a> of the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> with the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> <a href="/facts/Decidability_(logic)/17SHPMo0">decidable</a>?</li> <li>The universality problem for <i>C</i>-free graphs: For which finite sets <i>C</i> of graphs does the class of <i>C</i>-free countable graphs have a universal member under strong embeddings?<a class="footnote-ref" id="fnref:161" href="#fn:161"><sup>161</sup></a></li> <li>The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?<a class="footnote-ref" id="fnref:162" href="#fn:162"><sup>162</sup></a></li> <li><a href="/facts/Vaught_conjecture/FUw0PMR6">Vaught conjecture</a>: the number of <a href="/facts/Countable_set/Wj4DmWPv">countable</a> models of a <a href="/facts/First-order_logic/lcISqWIg">first-order</a> <a href="/facts/Complete_theory/TfBN8Fnb">complete theory</a> in a countable <a href="/facts/Formal_language/crDTyP8q">language</a> is either finite, ℵ 0 {\displaystyle \aleph _{0}} , or 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} .</li></ul> <ul><li>Assume <i>K</i> is the class of models of a countable first order theory omitting countably many <a href="/facts/Type_(model_theory)/nuUHGFcg">types</a>. If <i>K</i> has a model of cardinality ℵ ω 1 {\displaystyle \aleph _{\omega _{1}}} does it have a model of cardinality continuum?<a class="footnote-ref" id="fnref:163" href="#fn:163"><sup>163</sup></a></li> <li>Do the <a href="/facts/Henson_graph/RcWdR5Qu">Henson graphs</a> have the <a href="/facts/Finite_model_property/qD8M6fkT">finite model property</a>?</li> <li>Does a finitely presented homogeneous structure for a finite relational language have finitely many <a href="/facts/Reduct/RMh4Yvuu">reducts</a>?</li> <li>Does there exist an <a href="/facts/O-minimal/DoWoxEts">o-minimal</a> first order theory with a trans-exponential (rapid growth) function?</li> <li>If the class of atomic models of a complete first order theory is <a href="/facts/Categorical_(model_theory)/gTmQNu12">categorical</a> in the ℵ n {\displaystyle \aleph _{n}} , is it categorical in every cardinal?<a class="footnote-ref" id="fnref:164" href="#fn:164"><sup>164</sup></a><a class="footnote-ref" id="fnref:165" href="#fn:165"><sup>165</sup></a></li> <li>Is every infinite, minimal field of characteristic zero <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed</a>? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)</li> <li>Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?<a class="footnote-ref" id="fnref:166" href="#fn:166"><sup>166</sup></a></li> <li>Is the theory of the field of Laurent series over Z p {\displaystyle \mathbb {Z} _{p}} <a href="/facts/Decidability_(logic)/17SHPMo0">decidable</a>? of the field of polynomials over C {\displaystyle \mathbb {C} } ?</li> <li>Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?<a class="footnote-ref" id="fnref:167" href="#fn:167"><sup>167</sup></a></li></ul> <ul><li>Determine the structure of Keisler's order.<a class="footnote-ref" id="fnref:168" href="#fn:168"><sup>168</sup></a><a class="footnote-ref" id="fnref:169" href="#fn:169"><sup>169</sup></a></li></ul> <h3>Probability theory</h3> <p class="note">Main article: <a href="/facts/Probability_theory/mFBn51rE">Probability theory</a></p> <ul><li>Ibragimov–Iosifescu conjecture for φ-mixing sequences</li></ul> <h3>Number theory</h3> <p class="note">Main page: Category:Unsolved problems in number theory</p> <p class="note">See also: <a href="/facts/Number_theory/zhoa7zrc">Number theory</a></p> <h4>General</h4> <ul><li><a href="/facts/Special_values_of_L-functions/enVGIRfP">Beilinson's conjectures</a></li> <li><a href="/facts/Brocard%27s_problem/yf224Jby">Brocard's problem</a>: are there any integer solutions to n ! + 1 = m 2 {\displaystyle n!+1=m^{2}} other than n = 4 , 5 , 7 {\displaystyle n=4,5,7} ?</li> <li><a href="/facts/B%C3%BCchi%27s_problem/BgEIIR8m">Büchi's problem</a> on sufficiently large sequences of square numbers with constant second difference.</li> <li><a href="/facts/Carmichael%27s_totient_function_conjecture/ctFTcWKB">Carmichael's totient function conjecture</a>: do all values of <a href="/facts/Euler%27s_totient_function/mIQwXYzj">Euler's totient function</a> have <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a> greater than 1 {\displaystyle 1} ?</li> <li><a href="/facts/Casas-Alvero_conjecture/z4Ukf0fp">Casas-Alvero conjecture</a>: if a polynomial of degree d {\displaystyle d} defined over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K {\displaystyle K} of <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> 0 {\displaystyle 0} has a factor in common with its first through d − 1 {\displaystyle d-1} -th derivative, then must f {\displaystyle f} be the d {\displaystyle d} -th power of a linear polynomial?</li> <li><a href="/facts/Aliquot_sequence/XYI13Wne">Catalan–Dickson conjecture on aliquot sequences</a>: no <a href="/facts/Aliquot_sequence/XYI13Wne">aliquot sequences</a> are infinite but non-repeating.</li> <li><a href="/facts/Erd%C5%91s%E2%80%93Ulam_problem/ptXjS5qt">Erdős–Ulam problem</a>: is there a <a href="/facts/Dense_set/nPT9Yxao">dense set</a> of points in the plane all at rational distances from one-another?</li> <li><a href="/facts/Van_der_Corput%27s_method/BeVwy9R9">Exponent pair conjecture</a>: for all ε > 0 {\displaystyle \varepsilon >0} , is the pair ( ε , 1 / 2 + ε ) {\displaystyle (\varepsilon ,1/2+\varepsilon )} an <a href="/facts/Van_der_Corput%27s_method/BeVwy9R9">exponent pair</a>?</li> <li>The <a href="/facts/Gauss_circle_problem/L2S8dXdj">Gauss circle problem</a>: how far can the number of integer points in a circle centered at the origin be from the area of the circle?</li> <li><a href="/facts/Grand_Riemann_hypothesis/2AT9mrYl">Grand Riemann hypothesis</a>: do the nontrivial zeros of all <a href="/facts/Automorphic_L-function/fUEukywM">automorphic L-functions</a> lie on the critical line 1 / 2 + i t {\displaystyle 1/2+it} with real t {\displaystyle t} ? <ul><li><a href="/facts/Generalized_Riemann_hypothesis/17nLmBbu">Generalized Riemann hypothesis</a>: do the nontrivial zeros of all <a href="/facts/Dirichlet_L-function/my8hc6Qy">Dirichlet L-functions</a> lie on the critical line 1 / 2 + i t {\displaystyle 1/2+it} with real t {\displaystyle t} ? <ul><li><a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a>: do the nontrivial zeros of the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> lie on the critical line 1 / 2 + i t {\displaystyle 1/2+it} with real t {\displaystyle t} ?</li></ul></li></ul></li> <li><a href="/facts/Grimm%27s_conjecture/vTekPcnK">Grimm's conjecture</a>: each element of a set of consecutive <a href="/facts/Composite_number/SPtW9ftC">composite numbers</a> can be assigned a distinct <a href="/facts/Prime_number/L20FLkgn">prime number</a> that divides it.</li> <li><a href="/facts/Hall%27s_conjecture/38e462uo">Hall's conjecture</a>: for any ε > 0 {\displaystyle \varepsilon >0} , there is some constant c ( ε ) {\displaystyle c(\varepsilon )} such that either y 2 = x 3 {\displaystyle y^{2}=x^{3}} or | y 2 − x 3 | > c ( ε ) x 1 / 2 − ε {\displaystyle |y^{2}-x^{3}|>c(\varepsilon )x^{1/2-\varepsilon }} .</li> <li><a href="/facts/Hardy%E2%80%93Littlewood_zeta_function_conjectures/5JlCIvJn">Hardy–Littlewood zeta function conjectures</a></li> <li><a href="/facts/Hilbert%E2%80%93P%C3%B3lya_conjecture/HuNqSd7Q">Hilbert–Pólya conjecture</a>: the nontrivial zeros of the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> correspond to <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of a <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a>.</li> <li><a href="/facts/Hilbert%27s_eleventh_problem/3PmsWL77">Hilbert's eleventh problem</a>: classify <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a> over <a href="/facts/Algebraic_number_field/e0K0TXSh">algebraic number fields</a>.</li> <li><a href="/facts/Hilbert%27s_ninth_problem/7xURjXYE">Hilbert's ninth problem</a>: find the most general <a href="/facts/Reciprocity_law/s4YLAgn5">reciprocity law</a> for the <a href="/facts/Hilbert_symbol/KcQ6p12r">norm residues</a> of k {\displaystyle k} -th order in a general <a href="/facts/Algebraic_number_field/e0K0TXSh">algebraic number field</a>, where k {\displaystyle k} is a power of a prime.</li> <li><a href="/facts/Hilbert%27s_twelfth_problem/9II5Tu28">Hilbert's twelfth problem</a>: extend the <a href="/facts/Kronecker%E2%80%93Weber_theorem/U3pq1GAq">Kronecker–Weber theorem</a> on <a href="/facts/Abelian_extension/vIdW4Hnv">Abelian extensions</a> of Q {\displaystyle \mathbb {Q} } to any base number field.</li> <li>Keating–Snaith conjecture concerning the asymptotics of an integral involving the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a><a class="footnote-ref" id="fnref:170" href="#fn:170"><sup>170</sup></a></li> <li><a href="/facts/Lehmer%27s_totient_problem/nb0SIjT8">Lehmer's totient problem</a>: if ϕ ( n ) {\displaystyle \phi (n)} divides n − 1 {\displaystyle n-1} , must n {\displaystyle n} be prime?</li> <li><a href="/facts/Leopoldt%27s_conjecture/LUAUTl1v">Leopoldt's conjecture</a>: a <a href="/facts/P-adic_number/SDgDbkLF">p-adic</a> analogue of the <a href="/facts/Dirichlet%27s_unit_theorem/IQDk3QAR">regulator</a> of an <a href="/facts/Algebraic_number_field/e0K0TXSh">algebraic number field</a> does not vanish.</li> <li><a href="/facts/Lindel%C3%B6f_hypothesis/5Hfcmrxs">Lindelöf hypothesis</a> that for all ε > 0 {\displaystyle \varepsilon >0} , ζ ( 1 / 2 + i t ) = o ( t ε ) {\displaystyle \zeta (1/2+it)=o(t^{\varepsilon })} <ul><li>The <a href="/facts/Bombieri%E2%80%93Vinogradov_theorem/x83MjQoK">density hypothesis</a> for zeroes of the Riemann zeta function</li></ul></li> <li><a href="/facts/Littlewood_conjecture/e4AVb8QU">Littlewood conjecture</a>: for any two real numbers α , 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} , where ‖ x ‖ {\displaystyle \Vert x\Vert } is the distance from x {\displaystyle x} to the nearest integer.</li> <li><a href="/facts/Mahler%27s_3/2_problem/6LjeghDQ">Mahler's 3/2 problem</a> that no real number x {\displaystyle x} has the property that the fractional parts of x ( 3 / 2 ) n {\displaystyle x(3/2)^{n}} are less than 1 / 2 {\displaystyle 1/2} for all positive integers n {\displaystyle n} .</li> <li><a href="/facts/Montgomery%27s_pair_correlation_conjecture/6Q6RfeS7">Montgomery's pair correlation conjecture</a>: the normalized pair <a href="/facts/Correlation_function/8mmhdp8J">correlation function</a> between pairs of zeros of the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> is the same as the pair correlation function of <a href="/facts/Random_matrix/Yqo1lwFW">random Hermitian matrices</a>.</li> <li><a href="/facts/N_conjecture/B9Q264fh"><i>n</i> conjecture</a>: a generalization of the <i>abc</i> conjecture to more than three integers. <ul><li><a href="/facts/Abc_conjecture/yJeHld4Q"><i>abc</i> conjecture</a>: for any ε > 0 {\displaystyle \varepsilon >0} , rad ( a b c ) 1 + ε < c {\displaystyle \operatorname {rad} (abc)^{1+\varepsilon }<c} is true for only finitely many positive a , b , c {\displaystyle a,b,c} such that a + b = c {\displaystyle a+b=c} .</li> <li><a href="/facts/Szpiro%27s_conjecture/LqYE6vK5">Szpiro's conjecture</a>: for any ε > 0 {\displaystyle \varepsilon >0} , there is some constant C ( ε ) {\displaystyle C(\varepsilon )} such that, for any elliptic curve E {\displaystyle E} defined over Q {\displaystyle \mathbb {Q} } with minimal discriminant Δ {\displaystyle \Delta } and conductor f {\displaystyle f} , we have | Δ | ≤ C ( ε ) ⋅ f 6 + ε {\displaystyle |\Delta |\leq C(\varepsilon )\cdot f^{6+\varepsilon }} .</li></ul></li> <li><a href="/facts/Newman%27s_conjecture/PmmLK66w">Newman's conjecture</a>: the <a href="/facts/Partition_function_(number_theory)/mBUDy5vU">partition function</a> satisfies any arbitrary congruence infinitely often.</li> <li><a href="/facts/Divisor_summatory_function/LC52uv0F">Piltz divisor problem</a> on bounding Δ k ( x ) = D k ( x ) − x P k ( log ( x ) ) {\displaystyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(\log(x))} <ul><li><a href="/facts/Divisor_summatory_function/LC52uv0F">Dirichlet's divisor problem</a>: the specific case of the Piltz divisor problem for k = 1 {\displaystyle k=1} </li></ul></li> <li><a href="/facts/Ramanujan%E2%80%93Petersson_conjecture/vfu3TYM6">Ramanujan–Petersson conjecture</a>: a number of related conjectures that are generalizations of the original conjecture.</li> <li><a href="/facts/Sato%E2%80%93Tate_conjecture/BK5wTDyU">Sato–Tate conjecture</a>: also a number of related conjectures that are generalizations of the original conjecture.</li> <li><a href="/facts/Scholz_conjecture/dv9IelN4">Scholz conjecture</a>: the length of the shortest <a href="/facts/Addition_chain/WlqPzv4v">addition chain</a> producing 2 n − 1 {\displaystyle 2^{n}-1} is at most n − 1 {\displaystyle n-1} plus the length of the shortest addition chain producing n {\displaystyle n} .</li> <li>Do <a href="/facts/Siegel_zero/2sSmBoxW">Siegel zeros</a> exist?</li> <li><a href="/facts/Singmaster%27s_conjecture/9lFQKQUT">Singmaster's conjecture</a>: is there a finite upper bound on the multiplicities of the entries greater than 1 in <a href="/facts/Pascal%27s_triangle/tb6BKXbc">Pascal's triangle</a>?<a class="footnote-ref" id="fnref:171" href="#fn:171"><sup>171</sup></a></li> <li><a href="/facts/Vojta%27s_conjecture/cwmLbggM">Vojta's conjecture</a> on <a href="/facts/Height_function/ByeW10AJ">heights</a> of points on <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic varieties</a> over <a href="/facts/Algebraic_number_field/e0K0TXSh">algebraic number fields</a>.</li></ul> <ul><li>Are there infinitely many <a href="/facts/Perfect_number/WyqWtApV">perfect numbers</a>?</li> <li>Do any <a href="/facts/Odd_perfect_number/WyqWtApV">odd perfect numbers</a> exist?</li> <li>Do <a href="/facts/Quasiperfect_number/3BCbnTUC">quasiperfect numbers</a> exist?</li> <li>Do any non-power of 2 <a href="/facts/Almost_perfect_number/7pzWByG0">almost perfect numbers</a> exist?</li> <li>Are there 65, 66, or 67 <a href="/facts/Idoneal_number/gzUAFh2K">idoneal numbers</a>?</li> <li>Are there any pairs of <a href="/facts/Amicable_numbers/Rx7QgbXI">amicable numbers</a> which have opposite parity?</li> <li>Are there any pairs of <a href="/facts/Betrothed_numbers/WY5BDz5h">betrothed numbers</a> which have same parity?</li> <li>Are there any pairs of <a href="/facts/Relatively_prime/bnCxCXxs">relatively prime</a> <a href="/facts/Amicable_numbers/Rx7QgbXI">amicable numbers</a>?</li> <li>Are there infinitely many <a href="/facts/Amicable_numbers/Rx7QgbXI">amicable numbers</a>?</li> <li>Are there infinitely many <a href="/facts/Betrothed_numbers/WY5BDz5h">betrothed numbers</a>?</li> <li>Are there infinitely many <a href="/facts/Giuga_number/99ughX9G">Giuga numbers</a>?</li> <li>Does every <a href="/facts/Rational_number/VJQmyY05">rational number</a> with an odd denominator have an <a href="/facts/Odd_greedy_expansion/9tN97QuR">odd greedy expansion</a>?</li> <li>Do any <a href="/facts/Lychrel_number/Uwr7jcUk">Lychrel numbers</a> exist?</li> <li>Do any odd <a href="/facts/Noncototient/kEUz454A">noncototients</a> exist?</li> <li>Do any odd <a href="/facts/Weird_number/mrL00Uak">weird numbers</a> exist?</li> <li>Do any <a href="/facts/Superperfect_number/TwyHfzEQ">(2, 5)-perfect numbers</a> exist?</li> <li>Do any <a href="/facts/Generalized_taxicab_number/Bdz7Pa9Q">Taxicab(5, 2, n)</a> exist for <i>n</i> > 1?</li> <li>Is there a <a href="/facts/Covering_system/sXppI4U9">covering system</a> with odd distinct moduli?<a class="footnote-ref" id="fnref:172" href="#fn:172"><sup>172</sup></a></li> <li>Is π {\displaystyle \pi } a <a href="/facts/Normal_number/nGb5cM0y">normal number</a> (i.e., is each digit 0–9 equally frequent)?<a class="footnote-ref" id="fnref:173" href="#fn:173"><sup>173</sup></a></li> <li>Are all <a href="/facts/Irrational_number/Psv19TET">irrational</a> <a href="/facts/Algebraic_number/MkH1PzTK">algebraic</a> numbers normal?</li> <li>Is 10 a <a href="/facts/Solitary_number/XkcztDLo">solitary number</a>?</li> <li>Can a 3×3 <a href="/facts/Magic_square/QGQkSKF2">magic square</a> be constructed from 9 distinct perfect square numbers?<a class="footnote-ref" id="fnref:174" href="#fn:174"><sup>174</sup></a></li> <li>Find the value of the <a href="/facts/De_Bruijn%E2%80%93Newman_constant/mm9vT7US">De Bruijn–Newman constant</a>.</li></ul> <h4>Additive number theory</h4> <p class="note">Main article: <a href="/facts/Additive_number_theory/43Fm5NFN">Additive number theory</a></p> <p class="note">See also: <a href="/facts/Problems_involving_arithmetic_progressions/k8FohnnP">Problems involving arithmetic progressions</a></p> <ul><li><a href="/facts/Erd%C5%91s_conjecture_on_arithmetic_progressions/IqSRK67D">Erdős conjecture on arithmetic progressions</a> that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long <a href="/facts/Arithmetic_progression/c4ew9Xoq">arithmetic progressions</a>.</li> <li><a href="/facts/Erd%C5%91s%E2%80%93Tur%C3%A1n_conjecture_on_additive_bases/MXQHYSxk">Erdős–Turán conjecture on additive bases</a>: if B {\displaystyle B} is an <a href="/facts/Additive_basis/LLXCG8Xr">additive basis</a> of order 2 {\displaystyle 2} , then the number of ways that positive integers n {\displaystyle n} can be expressed as the sum of two numbers in B {\displaystyle B} must tend to infinity as n {\displaystyle n} tends to infinity.</li> <li><a href="/facts/Gilbreath%27s_conjecture/JBavkEg2">Gilbreath's conjecture</a> on consecutive applications of the unsigned <a href="/facts/Finite_difference/aY1txFhj">forward difference</a> operator to the sequence of <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>.</li> <li><a href="/facts/Goldbach%27s_conjecture/967UhtUU">Goldbach's conjecture</a>: every even natural number greater than 2 {\displaystyle 2} is the sum of two <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>.</li> <li><a href="/facts/Lander,_Parkin,_and_Selfridge_conjecture/oYMIAomA">Lander, Parkin, and Selfridge conjecture</a>: if the sum of m {\displaystyle m} k {\displaystyle k} -th powers of positive integers is equal to a different sum of n {\displaystyle n} k {\displaystyle k} -th powers of positive integers, then m + n ≥ k {\displaystyle m+n\geq k} .</li> <li><a href="/facts/Lemoine%27s_conjecture/3moVR9vU">Lemoine's conjecture</a>: all odd integers greater than 5 {\displaystyle 5} can be represented as the sum of an odd <a href="/facts/Prime_number/L20FLkgn">prime number</a> and an even <a href="/facts/Semiprime/J5cNFs7T">semiprime</a>.</li> <li><a href="/facts/Minimum_overlap_problem/eEDJpDdd">Minimum overlap problem</a> of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set { 1 , … , 2 n } {\displaystyle \{1,\ldots ,2n\}} </li> <li><a href="/facts/Pollock%27s_conjectures/BpEdHtR6">Pollock's conjectures</a></li> <li>Does every nonnegative integer appear in <a href="/facts/Recam%C3%A1n%27s_sequence/NtRnBzBF">Recamán's sequence</a>?</li> <li><a href="/facts/Skolem_problem/cMWS80o8">Skolem problem</a>: can an algorithm determine if a <a href="/facts/Constant-recursive_sequence/U3vHagFE">constant-recursive sequence</a> contains a zero?</li> <li>The values of <i>g</i>(<i>k</i>) and <i>G</i>(<i>k</i>) in <a href="/facts/Waring%27s_problem/Qq5My8wy">Waring's problem</a></li></ul> <ul><li>Do the <a href="/facts/Ulam_number/pxt5U3W9">Ulam numbers</a> have a positive density?</li></ul> <ul><li>Determine growth rate of <i>r</i><i>k</i>(<i>N</i>) (see <a href="/facts/Szemer%C3%A9di%27s_theorem/6fjvXcEx">Szemerédi's theorem</a>)</li></ul> <h4>Algebraic number theory</h4> <p class="note">Main article: <a href="/facts/Algebraic_number_theory/LPzDT5CA">Algebraic number theory</a></p> <ul><li><a href="/facts/Class_number_problem/bIrLDjde">Class number problem</a>: are there infinitely many <a href="/facts/Class_number_problem/bIrLDjde">real quadratic number fields</a> with <a href="/facts/Unique_factorization/aYsX2swC">unique factorization</a>?</li> <li><a href="/facts/Fontaine%E2%80%93Mazur_conjecture/q6oxsNwh">Fontaine–Mazur conjecture</a>: actually numerous conjectures, all proposed by <a href="/facts/Jean-Marc_Fontaine/qCDAhaex">Jean-Marc Fontaine</a> and <a href="/facts/Barry_Mazur/4tGbAnCX">Barry Mazur</a>.</li> <li><a href="/facts/Gan%E2%80%93Gross%E2%80%93Prasad_conjecture/1umdjT2E">Gan–Gross–Prasad conjecture</a>: a <a href="/facts/Restricted_representation/rw7IN86T">restriction</a> problem in <a href="/facts/Representation_of_a_Lie_group/29R96ar2">representation theory of real or p-adic Lie groups</a>.</li> <li><a href="/facts/Greenberg%27s_conjectures/Sag30tQT">Greenberg's conjectures</a></li> <li><a href="/facts/Hermite%27s_problem/V0qP7Y22">Hermite's problem</a>: is it possible, for any natural number n {\displaystyle n} , to assign a sequence of <a href="/facts/Natural_number/u65og8JE">natural numbers</a> to each <a href="/facts/Real_number/R02gw5Pb">real number</a> such that the sequence for x {\displaystyle x} is eventually <a href="/facts/Periodic_sequence/eIJhDfJn">periodic</a> if and only if x {\displaystyle x} is <a href="/facts/Algebraic_number/MkH1PzTK">algebraic</a> of degree n {\displaystyle n} ?</li> <li><a href="/facts/Kummer%E2%80%93Vandiver_conjecture/0krv81SK">Kummer–Vandiver conjecture</a>: primes p {\displaystyle p} do not divide the <a href="/facts/Ideal_class_group/LLn5Pwj1">class number</a> of the maximal real <a href="/facts/Field_extension/L7yIDtyK">subfield</a> of the p {\displaystyle p} -th <a href="/facts/Cyclotomic_field/cGxMMEoR">cyclotomic field</a>.</li> <li>Lang and Trotter's conjecture on <a href="/facts/Supersingular_prime_(algebraic_number_theory)/mv5rXd4D">supersingular primes</a> that the number of <a href="/facts/Supersingular_prime_(algebraic_number_theory)/mv5rXd4D">supersingular primes</a> less than a constant X {\displaystyle X} is within a constant multiple of X / ln X {\displaystyle {\sqrt {X}}/\ln {X}} </li> <li><a href="/facts/Selberg%27s_1/4_conjecture/BNnf9yAf">Selberg's 1/4 conjecture</a>: the <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of the <a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a> on <a href="/facts/Maass_wave_form/ep0I4xyk">Maass wave forms</a> of <a href="/facts/Congruence_subgroup/V334rHwz">congruence subgroups</a> are at least 1 / 4 {\displaystyle 1/4} .</li> <li><a href="/facts/Stark_conjectures/0BWpBDbY">Stark conjectures</a> (including <a href="/facts/Brumer%E2%80%93Stark_conjecture/18tbmCIf">Brumer–Stark conjecture</a>)</li></ul> <ul><li>Characterize all algebraic number fields that have some <a href="/facts/Algebraic_number_field/e0K0TXSh">power basis</a>.</li></ul> <h4>Computational number theory</h4> <p class="note">Main article: <a href="/facts/Computational_number_theory/7Lu5N57C">Computational number theory</a></p> <ul><li>Can <a href="/facts/Integer_factorization/EjMCTz2t">integer factorization</a> be done in <a href="/facts/Polynomial_time/77T62gmf">polynomial time</a>?</li></ul> <h4>Diophantine approximation and transcendental number theory</h4> <p class="note">Further information: <a href="/facts/Diophantine_approximation/WGVMuWF4">Diophantine approximation</a> and <a href="/facts/Transcendental_number_theory/tjyLHm5G">Transcendental number theory</a></p> <ul><li><a href="/facts/Schanuel%27s_conjecture/QF3gsdbK">Schanuel's conjecture</a> on the <a href="/facts/Transcendence_degree/ubxWrbjJ">transcendence degree</a> of certain <a href="/facts/Field_extension/L7yIDtyK">field extensions</a> of the rational numbers.<a class="footnote-ref" id="fnref:175" href="#fn:175"><sup>175</sup></a> In particular: Are π {\displaystyle \pi } and e {\displaystyle e} <a href="/facts/Algebraic_independence/H8wV21eD">algebraically independent</a>? Which nontrivial combinations of <a href="/facts/Transcendental_number/IrB7JppM">transcendental numbers</a> (such as e + π , e π , π e , π π , e e {\displaystyle e+\pi ,e\pi ,\pi ^{e},\pi ^{\pi },e^{e}} ) are themselves transcendental?<a class="footnote-ref" id="fnref:176" href="#fn:176"><sup>176</sup></a><a class="footnote-ref" id="fnref:177" href="#fn:177"><sup>177</sup></a></li> <li>The <a href="/facts/Four_exponentials_conjecture/cYhTS9Xo">four exponentials conjecture</a>: the transcendence of at least one of four exponentials of combinations of irrationals<a class="footnote-ref" id="fnref:178" href="#fn:178"><sup>178</sup></a></li> <li>Are <a href="/facts/Euler%E2%80%93Mascheroni_constant/2kEgRvjC">Euler's constant</a> γ {\displaystyle \gamma } and <a href="/facts/Catalan%27s_constant/c9fNqG8j">Catalan's constant</a> G {\displaystyle G} irrational? Are they transcendental? Is <a href="/facts/Ap%C3%A9ry%27s_constant/59do2TbQ">Apéry's constant</a> ζ ( 3 ) {\displaystyle \zeta (3)} transcendental?<a class="footnote-ref" id="fnref:179" href="#fn:179"><sup>179</sup></a><a class="footnote-ref" id="fnref:180" href="#fn:180"><sup>180</sup></a></li> <li>Which transcendental numbers are <a href="/facts/Period_(algebraic_geometry)/zbskleex">(exponential) periods</a>?<a class="footnote-ref" id="fnref:181" href="#fn:181"><sup>181</sup></a></li> <li>How well can <a href="/facts/Quadratic_equation/9IMnkJAT">non-quadratic</a> irrational numbers be approximated? What is the <a href="/facts/Irrationality_measure/Ck8DKqpH">irrationality measure</a> of specific (suspected) transcendental numbers such as π {\displaystyle \pi } and γ {\displaystyle \gamma } ?<a class="footnote-ref" id="fnref:182" href="#fn:182"><sup>182</sup></a></li> <li>Which irrational numbers have <a href="/facts/Simple_continued_fraction/Wg1cMPxN">simple continued fraction</a> terms whose <a href="/facts/Geometric_mean/6Nf6uk1A">geometric mean</a> converges to <a href="/facts/Khinchin%27s_constant/Wb8JMqGK">Khinchin's constant</a>?<a class="footnote-ref" id="fnref:183" href="#fn:183"><sup>183</sup></a></li></ul> <h4>Diophantine equations</h4> <p class="note">Further information: <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equation</a></p> <ul><li><a href="/facts/Beal%27s_conjecture/cWWrfR5d">Beal's conjecture</a>: for all integral solutions to A x + B y = C z {\displaystyle A^{x}+B^{y}=C^{z}} where x , y , z > 2 {\displaystyle x,y,z>2} , all three numbers A , B , C {\displaystyle A,B,C} must share some prime factor.</li> <li><a href="/facts/Congruent_number_problem/BGL6BjHl">Congruent number problem</a> (a corollary to <a href="/facts/Birch_and_Swinnerton-Dyer_conjecture/Issh7Yfr">Birch and Swinnerton-Dyer conjecture</a>, per <a href="/facts/Tunnell%27s_theorem/veEoQNnE">Tunnell's theorem</a>): determine precisely what rational numbers are <a href="/facts/Congruent_number/BGL6BjHl">congruent numbers</a>.</li> <li>Erdős–Moser problem: is 1 1 + 2 1 = 3 1 {\displaystyle 1^{1}+2^{1}=3^{1}} the only solution to the <a href="/facts/Erd%C5%91s%E2%80%93Moser_equation/bfSj5VKd">Erdős–Moser equation</a>?</li> <li><a href="/facts/Erd%C5%91s%E2%80%93Straus_conjecture/PYozIHvA">Erdős–Straus conjecture</a>: for every n ≥ 2 {\displaystyle n\geq 2} , there are positive integers x , y , z {\displaystyle x,y,z} such that 4 / n = 1 / x + 1 / y + 1 / z {\displaystyle 4/n=1/x+1/y+1/z} .</li> <li><a href="/facts/Fermat%E2%80%93Catalan_conjecture/RxRccY0M">Fermat–Catalan conjecture</a>: there are finitely many distinct solutions ( a m , b n , c k ) {\displaystyle (a^{m},b^{n},c^{k})} to the equation a m + b n = c k {\displaystyle a^{m}+b^{n}=c^{k}} with a , b , c {\displaystyle a,b,c} being positive <a href="/facts/Coprime_integers/bnCxCXxs">coprime integers</a> and m , n , k {\displaystyle m,n,k} being positive integers satisfying 1 / m + 1 / n + 1 / k < 1 {\displaystyle 1/m+1/n+1/k<1} .</li> <li><a href="/facts/Goormaghtigh_conjecture/DIsT7oRr">Goormaghtigh conjecture</a> on solutions to ( x m − 1 ) / ( x − 1 ) = ( y n − 1 ) / ( y − 1 ) {\displaystyle (x^{m}-1)/(x-1)=(y^{n}-1)/(y-1)} where x > y > 1 {\displaystyle x>y>1} and m , n > 2 {\displaystyle m,n>2} .</li> <li>The <a href="/facts/Markov_number/FVrXQvxS">uniqueness conjecture for Markov numbers</a><a class="footnote-ref" id="fnref:184" href="#fn:184"><sup>184</sup></a> that every <a href="/facts/Markov_number/FVrXQvxS">Markov number</a> is the largest number in exactly one normalized solution to the Markov <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equation</a>.</li> <li><a href="/facts/Pillai%27s_conjecture/LShwXn13">Pillai's conjecture</a>: for any A , B , C {\displaystyle A,B,C} , the equation A x m − B y n = C {\displaystyle Ax^{m}-By^{n}=C} has finitely many solutions when m , n {\displaystyle m,n} are not both 2 {\displaystyle 2} .</li> <li>Which integers can be written as the <a href="/facts/Sums_of_three_cubes/Pmwg1CQq">sum of three perfect cubes</a>?<a class="footnote-ref" id="fnref:185" href="#fn:185"><sup>185</sup></a></li> <li>Can every integer be written as a sum of four perfect cubes?</li></ul> <h4>Prime numbers</h4> <p class="note">Main article: <a href="/facts/Prime_numbers/L20FLkgn">Prime numbers</a></p> <ul><li><a href="/facts/Agoh%E2%80%93Giuga_conjecture/mmjrQf10">Agoh–Giuga conjecture</a> on the <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli numbers</a> that p {\displaystyle p} is prime if and only if p B p − 1 ≡ − 1 ( mod p ) {\displaystyle pB_{p-1}\equiv -1{\pmod {p}}} </li> <li><a href="/facts/Agrawal%27s_conjecture/WzpQszey">Agrawal's conjecture</a> that given <a href="/facts/Coprime_integers/bnCxCXxs">coprime positive integers</a> n {\displaystyle n} and r {\displaystyle r} , if ( X − 1 ) n ≡ X n − 1 ( mod n , X r − 1 ) {\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,X^{r}-1}}} , then either n {\displaystyle n} is prime or n 2 ≡ 1 ( mod r ) {\displaystyle n^{2}\equiv 1{\pmod {r}}} </li> <li><a href="/facts/Artin%27s_conjecture_on_primitive_roots/DWwlUWmk">Artin's conjecture on primitive roots</a> that if an integer is neither a perfect square nor − 1 {\displaystyle -1} , then it is a <a href="/facts/Primitive_root_modulo_n/QS4DbIB0">primitive root</a> modulo infinitely many <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> p {\displaystyle p} </li> <li><a href="/facts/Brocard%27s_conjecture/f7cBSkp6">Brocard's conjecture</a>: there are always at least 4 {\displaystyle 4} <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> between consecutive squares of prime numbers, aside from 2 2 {\displaystyle 2^{2}} and 3 2 {\displaystyle 3^{2}} .</li> <li><a href="/facts/Bunyakovsky_conjecture/xsnfHIj7">Bunyakovsky conjecture</a>: if an integer-coefficient polynomial f {\displaystyle f} has a positive leading coefficient, is irreducible over the integers, and has no common factors over all f ( x ) {\displaystyle f(x)} where x {\displaystyle x} is a positive integer, then f ( x ) {\displaystyle f(x)} is prime infinitely often.</li> <li><a href="/facts/Catalan%27s_Mersenne_conjecture/nPgxjq4z">Catalan's Mersenne conjecture</a>: some <a href="/facts/Double_Mersenne_number/nPgxjq4z">Catalan–Mersenne number</a> is composite and thus all Catalan–Mersenne numbers are composite after some point.</li> <li><a href="/facts/Dickson%27s_conjecture/PEftYcoe">Dickson's conjecture</a>: for a finite set of linear forms a 1 + b 1 n , … , a k + b k n {\displaystyle a_{1}+b_{1}n,\ldots ,a_{k}+b_{k}n} with each b i ≥ 1 {\displaystyle b_{i}\geq 1} , there are infinitely many n {\displaystyle n} for which all forms are <a href="/facts/Prime_number/L20FLkgn">prime</a>, unless there is some <a href="/facts/Modular_arithmetic/0wtRa5dU">congruence</a> condition preventing it.</li> <li>Dubner's conjecture: every even number greater than 4208 {\displaystyle 4208} is the sum of two <a href="/facts/Prime_number/L20FLkgn">primes</a> which both have a <a href="/facts/Twin_prime/272VWQbc">twin</a>.</li> <li><a href="/facts/Elliott%E2%80%93Halberstam_conjecture/j3nDYgm1">Elliott–Halberstam conjecture</a> on the distribution of <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> in <a href="/facts/Arithmetic_progression/c4ew9Xoq">arithmetic progressions</a>.</li> <li><a href="/facts/Powerful_number/uhIIrxcR">Erdős–Mollin–Walsh conjecture</a>: no three consecutive numbers are all <a href="/facts/Powerful_number/uhIIrxcR">powerful</a>.</li> <li><a href="/facts/Feit%E2%80%93Thompson_conjecture/yWPlBusG">Feit–Thompson conjecture</a>: for all distinct <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> p {\displaystyle p} and q {\displaystyle q} , ( p q − 1 ) / ( p − 1 ) {\displaystyle (p^{q}-1)/(p-1)} does not divide ( q p − 1 ) / ( q − 1 ) {\displaystyle (q^{p}-1)/(q-1)} </li> <li>Fortune's conjecture that no <a href="/facts/Fortunate_number/g4UPuCXq">Fortunate number</a> is composite.</li> <li>The <a href="/facts/Gaussian_moat/QqYGTRIB">Gaussian moat</a> problem: is it possible to find an infinite sequence of distinct <a href="/facts/Gaussian_prime_number/zj82AhtL">Gaussian prime numbers</a> such that the difference between consecutive numbers in the sequence is bounded?</li> <li><a href="/facts/Gillies%27_conjecture/eCeIR3El">Gillies' conjecture</a> on the distribution of <a href="/facts/Prime_number/L20FLkgn">prime</a> divisors of <a href="/facts/Mersenne_prime/IyF3sBNf">Mersenne numbers</a>.</li> <li><a href="/facts/Landau%27s_problems/vqLTtmeR">Landau's problems</a> <ul><li><a href="/facts/Goldbach_conjecture/967UhtUU">Goldbach conjecture</a>: all even <a href="/facts/Natural_number/u65og8JE">natural numbers</a> greater than 2 {\displaystyle 2} are the sum of two <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>.</li> <li><a href="/facts/Legendre%27s_conjecture/emDSyngz">Legendre's conjecture</a>: for every positive integer n {\displaystyle n} , there is a prime between n 2 {\displaystyle n^{2}} and ( n + 1 ) 2 {\displaystyle (n+1)^{2}} .</li> <li><a href="/facts/Twin_prime/272VWQbc">Twin prime conjecture</a>: there are infinitely many <a href="/facts/Twin_prime/272VWQbc">twin primes</a>.</li> <li>Are there infinitely many primes of the form n 2 + 1 {\displaystyle n^{2}+1} ?</li></ul></li> <li>Problems associated to <a href="/facts/Linnik%27s_theorem/uHBtyx00">Linnik's theorem</a></li> <li><a href="/facts/Mersenne_conjectures/ANbu3f3r">New Mersenne conjecture</a>: for any odd <a href="/facts/Natural_number/u65og8JE">natural number</a> p {\displaystyle p} , if any two of the three conditions p = 2 k ± 1 {\displaystyle p=2^{k}\pm 1} or p = 4 k ± 3 {\displaystyle p=4^{k}\pm 3} , 2 p − 1 {\displaystyle 2^{p}-1} is prime, and ( 2 p + 1 ) / 3 {\displaystyle (2^{p}+1)/3} is prime are true, then the third condition is also true.</li> <li><a href="/facts/Polignac%27s_conjecture/x0ndzGkg">Polignac's conjecture</a>: for all positive even numbers n {\displaystyle n} , there are infinitely many <a href="/facts/Prime_gap/quWSv5NR">prime gaps</a> of size n {\displaystyle n} .</li> <li><a href="/facts/Schinzel%27s_hypothesis_H/6Sckht9T">Schinzel's hypothesis H</a> that for every finite collection { f 1 , … , f k } {\displaystyle \{f_{1},\ldots ,f_{k}\}} of nonconstant <a href="/facts/Irreducible_polynomial/PTeuKm4W">irreducible polynomials</a> over the integers with positive leading coefficients, either there are infinitely many positive integers n {\displaystyle n} for which f 1 ( n ) , … , f k ( n ) {\displaystyle f_{1}(n),\ldots ,f_{k}(n)} are all <a href="/facts/Prime_number/L20FLkgn">primes</a>, or there is some fixed divisor m > 1 {\displaystyle m>1} which, for all n {\displaystyle n} , divides some f i ( n ) {\displaystyle f_{i}(n)} .</li> <li><a href="/facts/Sierpi%C5%84ski_number/3t2DxTql">Selfridge's conjecture</a>: is 78,557 the lowest <a href="/facts/Sierpi%C5%84ski_number/3t2DxTql">Sierpiński number</a>?</li> <li>Does the <a href="/facts/Wolstenholme%27s_theorem/xlgz2T42">converse of Wolstenholme's theorem</a> hold for all natural numbers?</li> <li>Are all <a href="/facts/Euclid_number/7WjyjxSz">Euclid numbers</a> <a href="/facts/Square-free_integer/R3qR9hP1">square-free</a>?</li> <li>Are all <a href="/facts/Fermat_number/SJT8fN9m">Fermat numbers</a> <a href="/facts/Square-free_integer/R3qR9hP1">square-free</a>?</li> <li>Are all <a href="/facts/Mersenne_number/IyF3sBNf">Mersenne numbers</a> of prime index <a href="/facts/Square-free_integer/R3qR9hP1">square-free</a>?</li> <li>Are there any composite <i>c</i> satisfying 2<i>c</i> − 1 ≡ 1 (mod <i>c</i>2)?</li> <li>Are there any <a href="/facts/Wall%E2%80%93Sun%E2%80%93Sun_prime/or8YpE95">Wall–Sun–Sun primes</a>?</li> <li>Are there any <a href="/facts/Wieferich_prime/tkc913fM">Wieferich primes</a> in base 47?</li> <li>Are there infinitely many <a href="/facts/Balanced_prime/yM1kucNh">balanced primes</a>?</li> <li>Are there infinitely many Carol primes?</li> <li>Are there infinitely many <a href="/facts/Cluster_prime/bznfNRqK">cluster primes</a>?</li> <li>Are there infinitely many <a href="/facts/Cousin_prime/QCmofxKu">cousin primes</a>?</li> <li>Are there infinitely many <a href="/facts/Cullen_number/Wp6b8VqI">Cullen primes</a>?</li> <li>Are there infinitely many <a href="/facts/Euclid_number/7WjyjxSz">Euclid primes</a>?</li> <li>Are there infinitely many <a href="/facts/Fibonacci_prime/ksNk86FU">Fibonacci primes</a>?</li> <li>Are there infinitely many <a href="/facts/Euclid_number/7WjyjxSz">Kummer primes</a>?</li> <li>Are there infinitely many Kynea primes?</li> <li>Are there infinitely many <a href="/facts/Lucas_number/tVbP4yQr">Lucas primes</a>?</li> <li>Are there infinitely many <a href="/facts/Mersenne_prime/IyF3sBNf">Mersenne primes</a> (<a href="/facts/Lenstra%E2%80%93Pomerance%E2%80%93Wagstaff_conjecture/ANbu3f3r">Lenstra–Pomerance–Wagstaff conjecture</a>); equivalently, infinitely many even <a href="/facts/Perfect_number/WyqWtApV">perfect numbers</a>?</li> <li>Are there infinitely many <a href="/facts/Newman%E2%80%93Shanks%E2%80%93Williams_prime/QRN3DyP9">Newman–Shanks–Williams primes</a>?</li> <li>Are there infinitely many <a href="/facts/Palindromic_prime/CHyCLdCG">palindromic primes</a> to every base?</li> <li>Are there infinitely many <a href="/facts/Pell_number/VMu7fXmx">Pell primes</a>?</li> <li>Are there infinitely many <a href="/facts/Pierpont_prime/MGNnTvpJ">Pierpont primes</a>?</li> <li>Are there infinitely many <a href="/facts/Prime_quadruplet/8nvfkJhm">prime quadruplets</a>?</li> <li>Are there infinitely many <a href="/facts/Prime_triplet/6mfQA1wB">prime triplets</a>?</li> <li><a href="/facts/Regular_prime/mXfxktKK">Siegel's conjecture</a>: are there infinitely many regular primes, and if so is their <a href="/facts/Natural_density/WtMTUmMJ">natural density</a> as a subset of all primes e − 1 / 2 {\displaystyle e^{-1/2}} ?</li> <li>Are there infinitely many <a href="/facts/Sexy_prime/w8VJxpxy">sexy primes</a>?</li> <li>Are there infinitely many <a href="/facts/Safe_and_Sophie_Germain_primes/d4pApf8z">safe and Sophie Germain primes</a>?</li> <li>Are there infinitely many <a href="/facts/Wagstaff_prime/xYvN1n39">Wagstaff primes</a>?</li> <li>Are there infinitely many <a href="/facts/Wieferich_prime/tkc913fM">Wieferich primes</a>?</li> <li>Are there infinitely many <a href="/facts/Wilson_prime/wRygKfvP">Wilson primes</a>?</li> <li>Are there infinitely many <a href="/facts/Wolstenholme_prime/0wTQK7Ca">Wolstenholme primes</a>?</li> <li>Are there infinitely many <a href="/facts/Woodall_number/MEBXUUqe">Woodall primes</a>?</li> <li>Can a prime <i>p</i> satisfy 2 p − 1 ≡ 1 ( mod p 2 ) {\displaystyle 2^{p-1}\equiv 1{\pmod {p^{2}}}} and 3 p − 1 ≡ 1 ( mod p 2 ) {\displaystyle 3^{p-1}\equiv 1{\pmod {p^{2}}}} simultaneously?<a class="footnote-ref" id="fnref:186" href="#fn:186"><sup>186</sup></a></li> <li>Does every prime number appear in the <a href="/facts/Euclid%E2%80%93Mullin_sequence/E8z92wzL">Euclid–Mullin sequence</a>?</li> <li>What is the smallest <a href="/facts/Skewes%27s_number/1CbBmfvz">Skewes's number</a>?</li> <li>For any given integer <i>a</i> > 0, are there infinitely many <a href="/facts/Lucas%E2%80%93Wieferich_prime/tkc913fM">Lucas–Wieferich primes</a> associated with the pair (<i>a</i>, −1)? (Specially, when <i>a</i> = 1, this is the Fibonacci-Wieferich primes, and when <i>a</i> = 2, this is the Pell-Wieferich primes)</li> <li>For any given integer <i>a</i> > 0, are there infinitely many primes <i>p</i> such that <i>a</i><i>p</i> − 1 ≡ 1 (mod <i>p</i>2)?<a class="footnote-ref" id="fnref:187" href="#fn:187"><sup>187</sup></a></li> <li>For any given integer <i>b</i> which is not a perfect power and not of the form −4<i>k</i>4 for integer <i>k</i>, are there infinitely many <a href="/facts/Repunit/tX1fSElj">repunit</a> primes to base <i>b</i>?</li> <li>For any given integers k ≥ 1 , b ≥ 2 , c ≠ 0 {\displaystyle k\geq 1,b\geq 2,c\neq 0} , with gcd(<i>k</i>, <i>c</i>) = 1 and gcd(<i>b</i>, <i>c</i>) = 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)} with integer <i>n</i> ≥ 1?</li> <li>Is every <a href="/facts/Fermat_number/SJT8fN9m">Fermat number</a> 2 2 n + 1 {\displaystyle 2^{2^{n}}+1} composite for n > 4 {\displaystyle n>4} ?</li> <li>Is 509,203 the lowest <a href="/facts/Riesel_number/jw7VIXPq">Riesel number</a>?</li></ul> <h3>Set theory</h3> <p class="note">Main article: <a href="/facts/Set_theory/1ptfsvUP">Set theory</a></p> <p>Note: These conjectures are about <a href="/facts/Model_theory/vLWVV8sQ">models</a> of <a href="/facts/Zermelo-Frankel_set_theory/UYIsPbXw">Zermelo-Frankel set theory</a> with <a href="/facts/Axiom_of_choice/1Ura2HTh">choice</a>, and may not be able to be expressed in models of other set theories such as the various <a href="/facts/Constructive_set_theory/myLo3Mjt">constructive set theories</a> or <a href="/facts/Non-wellfounded_set_theory/RkmGMC70">non-wellfounded set theory</a>. </p> <ul><li>(<a href="/facts/W._Hugh_Woodin/gvqYIXWt">Woodin</a>) Does the <a href="/facts/Generalized_continuum_hypothesis/R0yGqHND">generalized continuum hypothesis</a> below a <a href="/facts/Strongly_compact_cardinal/qecAN4SL">strongly compact cardinal</a> imply the <a href="/facts/Generalized_continuum_hypothesis/R0yGqHND">generalized continuum hypothesis</a> everywhere?</li> <li>Does the <a href="/facts/Generalized_continuum_hypothesis/R0yGqHND">generalized continuum hypothesis</a> entail <a href="/facts/Diamondsuit/e2sVSVjl"> ♢ ( E cf ( λ ) λ + ) {\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})} </a> for every <a href="/facts/Singular_cardinal/wrGvPz2c">singular cardinal</a> λ {\displaystyle \lambda } ?</li> <li>Does the <a href="/facts/Generalized_continuum_hypothesis/R0yGqHND">generalized continuum hypothesis</a> imply the existence of an <a href="/facts/Suslin_tree/h1LrlFhB">ℵ2-Suslin tree</a>?</li> <li>If ℵω is a strong limit cardinal, is 2 ℵ ω < ℵ ω 1 {\displaystyle 2^{\aleph _{\omega }}<\aleph _{\omega _{1}}} (see <a href="/facts/Singular_cardinals_hypothesis/2TgTEJDy">Singular cardinals hypothesis</a>)? The best bound, ℵω4, was obtained by <a href="/facts/Saharon_Shelah/oI5WSK8p">Shelah</a> using his <a href="/facts/PCF_theory/o8FmPmRr">PCF theory</a>.</li> <li>The problem of finding the ultimate <a href="/facts/Core_model/jxMx1G1o">core model</a>, one that contains all <a href="/facts/Large_cardinal_property/jkSeV92D">large cardinals</a>.</li> <li><a href="/facts/W._Hugh_Woodin/gvqYIXWt">Woodin's</a> <a href="/facts/%CE%A9-logic/rQXujQth">Ω-conjecture</a>: if there is a <a href="/facts/Class_(set_theory)/F5EmDd9b">proper class</a> of <a href="/facts/Woodin_cardinal/DL2ldbfY">Woodin cardinals</a>, then <a href="/facts/%CE%A9-logic/rQXujQth">Ω-logic</a> satisfies an analogue of <a href="/facts/G%C3%B6del%27s_completeness_theorem/yq02MUyw">Gödel's completeness theorem</a>.</li> <li>Does the <a href="/facts/Consistency/IHkdnA9M">consistency</a> of the existence of a <a href="/facts/Strongly_compact_cardinal/qecAN4SL">strongly compact cardinal</a> imply the consistent existence of a <a href="/facts/Supercompact_cardinal/I9FN4T8T">supercompact cardinal</a>?</li> <li>Does there exist a <a href="/facts/J%C3%B3nsson_cardinal/jBVsYDWq">Jónsson algebra</a> on ℵω?</li> <li>Is OCA (the <a href="/facts/Open_coloring_axiom/WHL7kkvB">open coloring axiom</a>) consistent with 2 ℵ 0 > ℵ 2 {\displaystyle 2^{\aleph _{0}}>\aleph _{2}} ?</li> <li><a href="/facts/Reinhardt_cardinal/jC2zb0yt">Reinhardt cardinals</a>: Without assuming the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>, can a <a href="/facts/Reinhardt_cardinal/jC2zb0yt">nontrivial elementary embedding</a> <i>V</i>→<i>V</i> exist?</li></ul> <h3>Topology</h3> <p class="note">Main article: <a href="/facts/Topology/2EqW47cU">Topology</a></p> <ul><li><a href="/facts/Baum%E2%80%93Connes_conjecture/YHRAAG7z">Baum–Connes conjecture</a>: the <a href="/facts/Baum%E2%80%93Connes_conjecture/YHRAAG7z">assembly map</a> is an <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>.</li> <li><a href="/facts/Berge_knot/wgSbhTed">Berge conjecture</a> that the only <a href="/facts/Knot_(mathematics)/UMtKJ36m">knots</a> in the <a href="/facts/3-sphere/u67nImpY">3-sphere</a> which admit <a href="/facts/Lens_space/5hQ8dgA5">lens space</a> <a href="/facts/Dehn_surgery/1MYnUs6S">surgeries</a> are <a href="/facts/Berge_knot/wgSbhTed">Berge knots</a>.</li> <li><a href="/facts/Bing%E2%80%93Borsuk_conjecture/8ybNHTfn">Bing–Borsuk conjecture</a>: every n {\displaystyle n} -dimensional <a href="/facts/Homogeneous_space/YGebqcJA">homogeneous</a> <a href="/facts/Retraction_(topology)/477Gm67f">absolute neighborhood retract</a> is a <a href="/facts/Topological_manifold/eLKXMzzR">topological manifold</a>.</li> <li><a href="/facts/Borel_conjecture/hNYGPhKB">Borel conjecture</a>: <a href="/facts/Aspherical_space/M43BNIGF">aspherical</a> <a href="/facts/Closed_manifold/jMG6G6Bl">closed manifolds</a> are determined up to <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a> by their <a href="/facts/Fundamental_group/VyYtLqSP">fundamental groups</a>.</li> <li><a href="/facts/Halperin_conjecture/rIE0BPKB">Halperin conjecture</a> on rational <a href="/facts/Serre_spectral_sequence/9kAf3sYV">Serre spectral sequences</a> of certain <a href="/facts/Fibration/h51Cx7xU">fibrations</a>.</li> <li><a href="/facts/Hilbert%E2%80%93Smith_conjecture/v6Qbohvs">Hilbert–Smith conjecture</a>: if a <a href="/facts/Locally_compact_space/hesalT7q">locally compact</a> <a href="/facts/Topological_group/cAgNMMRU">topological group</a> has a <a href="/facts/Continuous_function/sKbl02pB">continuous</a>, <a href="/facts/Group_action/Vh9VtUUw">faithful group action</a> on a <a href="/facts/Topological_manifold/eLKXMzzR">topological manifold</a>, then the group must be a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>.</li> <li>Mazur's conjectures<a class="footnote-ref" id="fnref:188" href="#fn:188"><sup>188</sup></a></li> <li><a href="/facts/Novikov_conjecture/ADvc1bUD">Novikov conjecture</a> on the <a href="/facts/Homotopy/1nWrPxdj">homotopy invariance</a> of certain <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> in the <a href="/facts/Pontryagin_class/TIFceC5j">Pontryagin classes</a> of a <a href="/facts/Manifold/rXPERJtu">manifold</a>, arising from the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a>.</li> <li><a href="/facts/Quadrisecant/JqRoYNnk">Quadrisecants</a> of <a href="/facts/Wild_knot/t996suvy">wild knots</a>: it has been conjectured that wild knots always have infinitely many quadrisecants.<a class="footnote-ref" id="fnref:189" href="#fn:189"><sup>189</sup></a></li> <li><a href="/facts/Ravenel_conjectures/pypnnjsv">Telescope conjecture</a>: the last of <a href="/facts/Ravenel%27s_conjectures/pypnnjsv">Ravenel's conjectures</a> in <a href="/facts/Stable_homotopy_theory/wnedyf7K">stable homotopy theory</a> to be resolved.<a class="footnote-ref" id="fnref:190" href="#fn:190"><sup>190</sup></a></li> <li><a href="/facts/Unknotting_problem/NSxIY5C6">Unknotting problem</a>: can <a href="/facts/Unknot/VvgWgJEr">unknots</a> be recognized in <a href="/facts/Time_complexity/77T62gmf">polynomial time</a>?</li> <li><a href="/facts/Volume_conjecture/1yhMHQF4">Volume conjecture</a> relating <a href="/facts/Quantum_invariant/kdERXpeG">quantum invariants</a> of <a href="/facts/Knot_(mathematics)/UMtKJ36m">knots</a> to the <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a> of their <a href="/facts/Knot_complement/wAQ07H0L">knot complements</a>.</li> <li><a href="/facts/Whitehead_conjecture/5hV6jvmP">Whitehead conjecture</a>: every <a href="/facts/Connectedness/P5w33wdx">connected</a> <a href="/facts/CW_complex/rN7buLMo">subcomplex</a> of a two-dimensional <a href="/facts/Aspherical_space/M43BNIGF">aspherical</a> <a href="/facts/CW_complex/rN7buLMo">CW complex</a> is aspherical.</li> <li><a href="/facts/Zeeman_conjecture/3xUED26r">Zeeman conjecture</a>: given a finite <a href="/facts/Contractible_space/REXYj8uS">contractible</a> two-dimensional <a href="/facts/CW_complex/rN7buLMo">CW complex</a> K {\displaystyle K} , is the space K × [ 0 , 1 ] {\displaystyle K\times [0,1]} <a href="/facts/Collapse_(topology)/kXCsPpLk">collapsible</a>?</li></ul> <h2 id="problems-solved-since-1995">Problems solved since 1995</h2> <h3>Algebra</h3> <ul><li><a href="/facts/Uniform_boundedness_conjecture_for_rational_points/74tLlRwE">Mazur's conjecture B</a> (Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger, 2020)<a class="footnote-ref" id="fnref:191" href="#fn:191"><sup>191</sup></a></li> <li>Suita conjecture (Qi'an Guan and <a href="/facts/Xiangyu_Zhou/roVpbCgs">Xiangyu Zhou</a>, 2015) <a class="footnote-ref" id="fnref:192" href="#fn:192"><sup>192</sup></a></li> <li><a href="/facts/Torsion_conjecture/TyyUtQ4F">Torsion conjecture</a> (<a href="/facts/Lo%C3%AFc_Merel/9qxJyPuu">Loïc Merel</a>, 1996)<a class="footnote-ref" id="fnref:193" href="#fn:193"><sup>193</sup></a></li> <li><a href="/facts/Carlitz%E2%80%93Wan_conjecture/MAy8cn4E">Carlitz–Wan conjecture</a> (<a href="/facts/Hendrik_Lenstra/8XQ4Izlf">Hendrik Lenstra</a>, 1995)<a class="footnote-ref" id="fnref:194" href="#fn:194"><sup>194</sup></a></li> <li><a href="/facts/Serre%27s_multiplicity_conjectures/lIA8APPD">Serre's nonnegativity conjecture</a> (<a href="/facts/Ofer_Gabber/cp5SDafJ">Ofer Gabber</a>, 1995)</li></ul> <h3>Analysis</h3> <ul><li><a href="/facts/Kadison%E2%80%93Singer_problem/jVlagSuW">Kadison–Singer problem</a> (<a href="/facts/Adam_Marcus_(mathematician)/PwgjlGMr">Adam Marcus</a>, <a href="/facts/Daniel_Spielman/Gspcy7D0">Daniel Spielman</a> and <a href="/facts/Nikhil_Srivastava/DondKFN0">Nikhil Srivastava</a>, 2013)<a class="footnote-ref" id="fnref:195" href="#fn:195"><sup>195</sup></a><a class="footnote-ref" id="fnref:196" href="#fn:196"><sup>196</sup></a> (and the <a href="/facts/Hans_Georg_Feichtinger/m7dXLsBj">Feichtinger's conjecture</a>, Anderson's paving conjectures, Weaver's discrepancy theoretic K S r {\displaystyle KS_{r}} and K S r ′ {\displaystyle KS'_{r}} conjectures, Bourgain-Tzafriri conjecture and R ε {\displaystyle R_{\varepsilon }} -conjecture)</li> <li><a href="/facts/Ahlfors_measure_conjecture/CqVF1u5q">Ahlfors measure conjecture</a> (<a href="/facts/Ian_Agol/ozxjQ99k">Ian Agol</a>, 2004)<a class="footnote-ref" id="fnref:197" href="#fn:197"><sup>197</sup></a></li> <li><a href="/facts/Gradient_conjecture/6QFIjl8p">Gradient conjecture</a> (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)<a class="footnote-ref" id="fnref:198" href="#fn:198"><sup>198</sup></a></li></ul> <h3>Combinatorics</h3> <ul><li><a href="/facts/Erd%C5%91s_sumset_conjecture/z3buWCB9">Erdős sumset conjecture</a> (Joel Moreira, Florian Richter, Donald Robertson, 2018)<a class="footnote-ref" id="fnref:199" href="#fn:199"><sup>199</sup></a></li> <li><a href="/facts/Simplicial_sphere/zbGaCRwj">McMullen's g-conjecture</a> on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)<a class="footnote-ref" id="fnref:200" href="#fn:200"><sup>200</sup></a><a class="footnote-ref" id="fnref:201" href="#fn:201"><sup>201</sup></a></li> <li><a href="/facts/Hirsch_conjecture/MqC0SHcj">Hirsch conjecture</a> (<a href="/facts/Francisco_Santos_Leal/sb8aHd6d">Francisco Santos Leal</a>, 2010)<a class="footnote-ref" id="fnref:202" href="#fn:202"><sup>202</sup></a><a class="footnote-ref" id="fnref:203" href="#fn:203"><sup>203</sup></a></li> <li><a href="/facts/Ira_Gessel/jd5Xxa4f">Gessel's lattice path conjecture</a> (<a href="/facts/Manuel_Kauers/ffIQvP7N">Manuel Kauers</a>, <a href="/facts/Christoph_Koutschan/r3G4VFCQ">Christoph Koutschan</a>, and <a href="/facts/Doron_Zeilberger/M90TmhyY">Doron Zeilberger</a>, 2009)<a class="footnote-ref" id="fnref:204" href="#fn:204"><sup>204</sup></a></li> <li><a href="/facts/Stanley%E2%80%93Wilf_conjecture/pHDS7tHY">Stanley–Wilf conjecture</a> (<a href="/facts/G%C3%A1bor_Tardos/rKubPwsY">Gábor Tardos</a> and <a href="/facts/Adam_Marcus_(mathematician)/PwgjlGMr">Adam Marcus</a>, 2004)<a class="footnote-ref" id="fnref:205" href="#fn:205"><sup>205</sup></a> (and also the Alon–Friedgut conjecture)</li> <li><a href="/facts/Kemnitz%27s_conjecture/b77nLsBu">Kemnitz's conjecture</a> (<a href="/facts/Christian_Reiher/WkIKC1N4">Christian Reiher</a>, 2003, Carlos di Fiore, 2003)<a class="footnote-ref" id="fnref:206" href="#fn:206"><sup>206</sup></a></li> <li><a href="/facts/Cameron%E2%80%93Erd%C5%91s_conjecture/uhR4lF0H">Cameron–Erdős conjecture</a> (<a href="/facts/Ben_J._Green/eU3YvvgM">Ben J. Green</a>, 2003, Alexander Sapozhenko, 2003)<a class="footnote-ref" id="fnref:207" href="#fn:207"><sup>207</sup></a><a class="footnote-ref" id="fnref:208" href="#fn:208"><sup>208</sup></a></li></ul> <h3>Dynamical systems</h3> <ul><li><a href="/facts/Zimmer%27s_conjecture/0b1AWBAf">Zimmer's conjecture</a> (Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar, 2017)<a class="footnote-ref" id="fnref:209" href="#fn:209"><sup>209</sup></a></li> <li><a href="/facts/Painlev%C3%A9_conjecture/ICRxvM91">Painlevé conjecture</a> (Jinxin Xue, 2014)<a class="footnote-ref" id="fnref:210" href="#fn:210"><sup>210</sup></a><a class="footnote-ref" id="fnref:211" href="#fn:211"><sup>211</sup></a></li></ul> <h3>Game theory</h3> <ul><li>Existence of a non-terminating game of <a href="/facts/Beggar-my-neighbour/5tkLRCsE">beggar-my-neighbour</a> (Brayden Casella, 2024)<a class="footnote-ref" id="fnref:212" href="#fn:212"><sup>212</sup></a></li> <li>The <a href="/facts/Angel_problem/L207aEXQ">angel problem</a> (Various independent proofs, 2006)<a class="footnote-ref" id="fnref:213" href="#fn:213"><sup>213</sup></a><a class="footnote-ref" id="fnref:214" href="#fn:214"><sup>214</sup></a><a class="footnote-ref" id="fnref:215" href="#fn:215"><sup>215</sup></a><a class="footnote-ref" id="fnref:216" href="#fn:216"><sup>216</sup></a></li></ul> <h3>Geometry</h3> <h4>21st century</h4> <ul><li><a href="/facts/Einstein_problem/TKzzvTSx">Einstein problem</a> (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2024)<a class="footnote-ref" id="fnref:217" href="#fn:217"><sup>217</sup></a></li> <li>Maximal rank conjecture (Eric Larson, 2018)<a class="footnote-ref" id="fnref:218" href="#fn:218"><sup>218</sup></a></li> <li><a href="/facts/Weibel%27s_conjecture/og7omJLF">Weibel's conjecture</a> (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018)<a class="footnote-ref" id="fnref:219" href="#fn:219"><sup>219</sup></a></li> <li><a href="/facts/Yau%27s_conjecture/zBSMPlnN">Yau's conjecture</a> (<a href="/facts/Antoine_Song/5WA4LUvk">Antoine Song</a>, 2018)<a class="footnote-ref" id="fnref:220" href="#fn:220"><sup>220</sup></a><a class="footnote-ref" id="fnref:221" href="#fn:221"><sup>221</sup></a></li> <li><a href="/facts/Pentagonal_tiling/zAEA5rzw">Pentagonal tiling</a> (Michaël Rao, 2017)<a class="footnote-ref" id="fnref:222" href="#fn:222"><sup>222</sup></a></li> <li><a href="/facts/Willmore_conjecture/DJqRzSky">Willmore conjecture</a> (<a href="/facts/Fernando_Cod%C3%A1_Marques/YIGzq3h8">Fernando Codá Marques</a> and <a href="/facts/Andr%C3%A9_Neves/r4y644jP">André Neves</a>, 2012)<a class="footnote-ref" id="fnref:223" href="#fn:223"><sup>223</sup></a></li> <li><a href="/facts/Erd%C5%91s_distinct_distances_problem/81vRfEoA">Erdős distinct distances problem</a> (<a href="/facts/Larry_Guth/q5fLy7NX">Larry Guth</a>, <a href="/facts/Nets_Katz/K4uXYjhh">Nets Hawk Katz</a>, 2011)<a class="footnote-ref" id="fnref:224" href="#fn:224"><sup>224</sup></a></li> <li><a href="/facts/Squaring_the_plane/8LPIGM4t">Heterogeneous tiling conjecture (squaring the plane)</a> (Frederick V. Henle and James M. Henle, 2008)<a class="footnote-ref" id="fnref:225" href="#fn:225"><sup>225</sup></a></li> <li><a href="/facts/Tameness_conjecture/TsrXGtBT">Tameness conjecture</a> (<a href="/facts/Ian_Agol/ozxjQ99k">Ian Agol</a>, 2004)<a class="footnote-ref" id="fnref:226" href="#fn:226"><sup>226</sup></a></li> <li><a href="/facts/Ending_lamination_theorem/rVnmXdMb">Ending lamination theorem</a> (<a href="/facts/Jeffrey_Brock/lp13yWhH">Jeffrey F. Brock</a>, <a href="/facts/Richard_Canary/WasRWMIH">Richard D. Canary</a>, <a href="/facts/Yair_Minsky/uIlHcdFn">Yair N. Minsky</a>, 2004)<a class="footnote-ref" id="fnref:227" href="#fn:227"><sup>227</sup></a></li> <li><a href="/facts/Carpenter%27s_rule_problem/93RNqj4t">Carpenter's rule problem</a> (<a href="/facts/Robert_Connelly/IUxlRmup">Robert Connelly</a>, <a href="/facts/Erik_Demaine/qby54dWx">Erik Demaine</a>, Günter Rote, 2003)<a class="footnote-ref" id="fnref:228" href="#fn:228"><sup>228</sup></a></li> <li><a href="/facts/Lambda_g_conjecture/0n42e1Df">Lambda g conjecture</a> (Carel Faber and <a href="/facts/Rahul_Pandharipande/9FhB30g5">Rahul Pandharipande</a>, 2003)<a class="footnote-ref" id="fnref:229" href="#fn:229"><sup>229</sup></a></li> <li><a href="/facts/Nagata%27s_conjecture/nfKoeIDE">Nagata's conjecture</a> (Ivan Shestakov, Ualbai Umirbaev, 2003)<a class="footnote-ref" id="fnref:230" href="#fn:230"><sup>230</sup></a></li> <li><a href="/facts/Double_bubble_conjecture/e5UV9z7r">Double bubble conjecture</a> (<a href="/facts/Michael_Hutchings_(mathematician)/SkmAIg6H">Michael Hutchings</a>, <a href="/facts/Frank_Morgan_(mathematician)/VfSdkRLl">Frank Morgan</a>, Manuel Ritoré, Antonio Ros, 2002)<a class="footnote-ref" id="fnref:231" href="#fn:231"><sup>231</sup></a></li></ul> <h4>20th century</h4> <ul><li><a href="/facts/Honeycomb_conjecture/g45j09Io">Honeycomb conjecture</a> (<a href="/facts/Thomas_Callister_Hales/c0cWXvC0">Thomas Callister Hales</a>, 1999)<a class="footnote-ref" id="fnref:232" href="#fn:232"><sup>232</sup></a></li> <li><a href="/facts/Lange%27s_conjecture/ev2YD5vN">Lange's conjecture</a> (<a href="/facts/Montserrat_Teixidor_i_Bigas/I0BqmnzP">Montserrat Teixidor i Bigas</a> and Barbara Russo, 1999)<a class="footnote-ref" id="fnref:233" href="#fn:233"><sup>233</sup></a></li> <li><a href="/facts/Bogomolov_conjecture/STNEIuyy">Bogomolov conjecture</a> (<a href="/facts/Emmanuel_Ullmo/QY3p6B9s">Emmanuel Ullmo</a>, 1998, <a href="/facts/Shou-Wu_Zhang/mwkykgLj">Shou-Wu Zhang</a>, 1998)<a class="footnote-ref" id="fnref:234" href="#fn:234"><sup>234</sup></a><a class="footnote-ref" id="fnref:235" href="#fn:235"><sup>235</sup></a></li> <li><a href="/facts/Kepler_conjecture/5cFNDqwA">Kepler conjecture</a> (Samuel Ferguson, <a href="/facts/Thomas_Callister_Hales/c0cWXvC0">Thomas Callister Hales</a>, 1998)<a class="footnote-ref" id="fnref:236" href="#fn:236"><sup>236</sup></a></li> <li><a href="/facts/Dodecahedral_conjecture/4BDjCjFN">Dodecahedral conjecture</a> (<a href="/facts/Thomas_Callister_Hales/c0cWXvC0">Thomas Callister Hales</a>, Sean McLaughlin, 1998)<a class="footnote-ref" id="fnref:237" href="#fn:237"><sup>237</sup></a></li></ul> <h3>Graph theory</h3> <ul><li><a href="/facts/Kahn%E2%80%93Kalai_conjecture/YKo7v1Q6">Kahn–Kalai conjecture</a> (<a href="/facts/Jinyoung_Park_(mathematician)/LwVzzucS">Jinyoung Park</a> and Huy Tuan Pham, 2022)<a class="footnote-ref" id="fnref:238" href="#fn:238"><sup>238</sup></a></li> <li><a href="/facts/Blankenship%E2%80%93Oporowski_conjecture/2hz3yIP7">Blankenship–Oporowski conjecture</a> on the book thickness of subdivisions (<a href="/facts/Vida_Dujmovi%C4%87/J2uuBFdx">Vida Dujmović</a>, <a href="/facts/David_Eppstein/hIjTU6nC">David Eppstein</a>, Robert Hickingbotham, <a href="/facts/Pat_Morin/oCF25uLk">Pat Morin</a>, and <a href="/facts/David_Wood_(mathematician)/LSkm0hV6">David Wood</a>, 2021)<a class="footnote-ref" id="fnref:239" href="#fn:239"><sup>239</sup></a></li> <li><a href="/facts/Graceful_labeling/Do7bxMnc">Ringel's conjecture</a> that the complete graph K 2 n + 1 {\displaystyle K_{2n+1}} can be decomposed into 2 n + 1 {\displaystyle 2n+1} copies of any tree with n {\displaystyle n} edges (Richard Montgomery, <a href="/facts/Benny_Sudakov/gYyxp5ym">Benny Sudakov</a>, Alexey Pokrovskiy, 2020)<a class="footnote-ref" id="fnref:240" href="#fn:240"><sup>240</sup></a><a class="footnote-ref" id="fnref:241" href="#fn:241"><sup>241</sup></a></li> <li>Disproof of <a href="/facts/Hedetniemi%27s_conjecture/FuvrM96u">Hedetniemi's conjecture</a> on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)<a class="footnote-ref" id="fnref:242" href="#fn:242"><sup>242</sup></a></li> <li><a href="/facts/Kelmans%E2%80%93Seymour_conjecture/MFMVBRHT">Kelmans–Seymour conjecture</a> (Dawei He, Yan Wang, and Xingxing Yu, 2020)<a class="footnote-ref" id="fnref:243" href="#fn:243"><sup>243</sup></a><a class="footnote-ref" id="fnref:244" href="#fn:244"><sup>244</sup></a><a class="footnote-ref" id="fnref:245" href="#fn:245"><sup>245</sup></a><a class="footnote-ref" id="fnref:246" href="#fn:246"><sup>246</sup></a></li> <li><a href="/facts/Goldberg%E2%80%93Seymour_conjecture/xcCgy0V9">Goldberg–Seymour conjecture</a> (Guantao Chen, Guangming Jing, and Wenan Zang, 2019)<a class="footnote-ref" id="fnref:247" href="#fn:247"><sup>247</sup></a></li> <li><a href="/facts/Babai%27s_problem/BkSwHgld">Babai's problem</a> (Alireza Abdollahi, Maysam Zallaghi, 2015)<a class="footnote-ref" id="fnref:248" href="#fn:248"><sup>248</sup></a></li> <li><a href="/facts/Alspach%27s_conjecture/mnRHCLkY">Alspach's conjecture</a> (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)</li> <li><a href="/facts/Alon%E2%80%93Saks%E2%80%93Seymour_conjecture/fjsbXxsG">Alon–Saks–Seymour conjecture</a> (Hao Huang, <a href="/facts/Benny_Sudakov/gYyxp5ym">Benny Sudakov</a>, 2012)</li> <li><a href="/facts/Read%27s_conjecture/gQsUHaab">Read–Hoggar conjecture</a> (<a href="/facts/June_Huh/Kq61rldd">June Huh</a>, 2009)<a class="footnote-ref" id="fnref:249" href="#fn:249"><sup>249</sup></a></li> <li><a href="/facts/Scheinerman%27s_conjecture/EdhsUr2b">Scheinerman's conjecture</a> (Jeremie Chalopin and Daniel Gonçalves, 2009)<a class="footnote-ref" id="fnref:250" href="#fn:250"><sup>250</sup></a></li> <li><a href="/facts/Erd%C5%91s%E2%80%93Menger_conjecture/NlHduKXm">Erdős–Menger conjecture</a> (<a href="/facts/Ron_Aharoni/oPYBXbL6">Ron Aharoni</a>, Eli Berger 2007)<a class="footnote-ref" id="fnref:251" href="#fn:251"><sup>251</sup></a></li> <li><a href="/facts/Road_coloring_conjecture/gMBl1l1D">Road coloring conjecture</a> (<a href="/facts/Avraham_Trahtman/FhMMkonG">Avraham Trahtman</a>, 2007)<a class="footnote-ref" id="fnref:252" href="#fn:252"><sup>252</sup></a></li> <li><a href="/facts/Robertson%E2%80%93Seymour_theorem/J7g4CJ74">Robertson–Seymour theorem</a> (<a href="/facts/Neil_Robertson_(mathematician)/nC5FXjur">Neil Robertson</a>, <a href="/facts/Paul_Seymour_(mathematician)/AJp2GgAC">Paul Seymour</a>, 2004)<a class="footnote-ref" id="fnref:253" href="#fn:253"><sup>253</sup></a></li> <li><a href="/facts/Strong_perfect_graph_conjecture/XcIR3USE">Strong perfect graph conjecture</a> (<a href="/facts/Maria_Chudnovsky/MrP7qyTT">Maria Chudnovsky</a>, <a href="/facts/Neil_Robertson_(mathematician)/nC5FXjur">Neil Robertson</a>, <a href="/facts/Paul_Seymour_(mathematician)/AJp2GgAC">Paul Seymour</a> and <a href="/facts/Robin_Thomas_(mathematician)/A3fS9arb">Robin Thomas</a>, 2002)<a class="footnote-ref" id="fnref:254" href="#fn:254"><sup>254</sup></a></li> <li><a href="/facts/Toida%27s_conjecture/KljCvwK2">Toida's conjecture</a> (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001)<a class="footnote-ref" id="fnref:255" href="#fn:255"><sup>255</sup></a></li> <li>Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996)<a class="footnote-ref" id="fnref:256" href="#fn:256"><sup>256</sup></a></li></ul> <h3>Group theory</h3> <ul><li><a href="/facts/Hanna_Neumann_conjecture/Kdrg03oa">Hanna Neumann conjecture</a> (Joel Friedman, 2011, Igor Mineyev, 2011)<a class="footnote-ref" id="fnref:257" href="#fn:257"><sup>257</sup></a><a class="footnote-ref" id="fnref:258" href="#fn:258"><sup>258</sup></a></li> <li><a href="/facts/Density_theorem_for_Kleinian_groups/71sde7T5">Density theorem</a> (Hossein Namazi, Juan Souto, 2010)<a class="footnote-ref" id="fnref:259" href="#fn:259"><sup>259</sup></a></li> <li>Full <a href="/facts/Classification_of_finite_simple_groups/nMwLJf1K">classification of finite simple groups</a> (<a href="/facts/Koichiro_Harada/lh5msuwS">Koichiro Harada</a>, <a href="/facts/Ronald_Solomon/vh2Kayk5">Ronald Solomon</a>, 2008)</li></ul> <h3>Number theory</h3> <h4>21st century</h4> <ul><li><a href="/facts/Andr%C3%A9%E2%80%93Oort_conjecture/CjJ77kwc">André–Oort conjecture</a> (<a href="/facts/Jonathan_Pila/0Rpy11Kx">Jonathan Pila</a>, Ananth Shankar, <a href="/facts/Jacob_Tsimerman/1dAtNbYx">Jacob Tsimerman</a>, 2021)<a class="footnote-ref" id="fnref:260" href="#fn:260"><sup>260</sup></a></li> <li><a href="/facts/Duffin%E2%80%93Schaeffer_conjecture/NKC2mK7C">Duffin-Schaeffer conjecture</a> (<a href="/facts/Dimitris_Koukoulopoulos/qHvpT7Dd">Dimitris Koukoulopoulos</a>, <a href="/facts/James_Maynard_(mathematician)/3u2VgW1h">James Maynard</a>, 2019)</li> <li><a href="/facts/Vinogradov%27s_mean-value_theorem/beILYTQP">Main conjecture in Vinogradov's mean-value theorem</a> (<a href="/facts/Jean_Bourgain/Va4EpK8V">Jean Bourgain</a>, Ciprian Demeter, <a href="/facts/Larry_Guth/q5fLy7NX">Larry Guth</a>, 2015)<a class="footnote-ref" id="fnref:261" href="#fn:261"><sup>261</sup></a></li> <li><a href="/facts/Goldbach%27s_weak_conjecture/yCDWbzlK">Goldbach's weak conjecture</a> (<a href="/facts/Harald_Helfgott/CurPobx2">Harald Helfgott</a>, 2013)<a class="footnote-ref" id="fnref:262" href="#fn:262"><sup>262</sup></a><a class="footnote-ref" id="fnref:263" href="#fn:263"><sup>263</sup></a><a class="footnote-ref" id="fnref:264" href="#fn:264"><sup>264</sup></a></li> <li><a href="/facts/Prime_gap/quWSv5NR">Existence of bounded gaps between arbitrarily large primes</a> (<a href="/facts/Yitang_Zhang/Xd2tT5uP">Yitang Zhang</a>, <a href="/facts/Polymath_Project/LGj7RXQ9">Polymath8</a>, <a href="/facts/James_Maynard_(mathematician)/3u2VgW1h">James Maynard</a>, 2013)<a class="footnote-ref" id="fnref:265" href="#fn:265"><sup>265</sup></a><a class="footnote-ref" id="fnref:266" href="#fn:266"><sup>266</sup></a><a class="footnote-ref" id="fnref:267" href="#fn:267"><sup>267</sup></a></li> <li><a href="/facts/Sidon_sequence/XNaVvett">Sidon set problem</a> (Javier Cilleruelo, <a href="/facts/Imre_Z._Ruzsa/pRszA5IK">Imre Z. Ruzsa</a>, and Carlos Vinuesa, 2010)<a class="footnote-ref" id="fnref:268" href="#fn:268"><sup>268</sup></a></li> <li><a href="/facts/Serre%27s_modularity_conjecture/8pYSFbHc">Serre's modularity conjecture</a> (<a href="/facts/Chandrashekhar_Khare/Ys2XnJVa">Chandrashekhar Khare</a> and <a href="/facts/Jean-Pierre_Wintenberger/rpFu37a1">Jean-Pierre Wintenberger</a>, 2008)<a class="footnote-ref" id="fnref:269" href="#fn:269"><sup>269</sup></a><a class="footnote-ref" id="fnref:270" href="#fn:270"><sup>270</sup></a><a class="footnote-ref" id="fnref:271" href="#fn:271"><sup>271</sup></a></li> <li><a href="/facts/Green%E2%80%93Tao_theorem/kGTx52Vp">Green–Tao theorem</a> (<a href="/facts/Ben_J._Green/eU3YvvgM">Ben J. Green</a> and <a href="/facts/Terence_Tao/Do68JKn6">Terence Tao</a>, 2004)<a class="footnote-ref" id="fnref:272" href="#fn:272"><sup>272</sup></a></li> <li><a href="/facts/Mih%C4%83ilescu%27s_theorem/LShwXn13">Catalan's conjecture</a> (<a href="/facts/Preda_Mih%C4%83ilescu/yIuayGhd">Preda Mihăilescu</a>, 2002)<a class="footnote-ref" id="fnref:273" href="#fn:273"><sup>273</sup></a></li> <li><a href="/facts/Erd%C5%91s%E2%80%93Graham_problem/jz6ebkqT">Erdős–Graham problem</a> (<a href="/facts/Ernest_S._Croot_III/huLgNQxq">Ernest S. Croot III</a>, 2000)<a class="footnote-ref" id="fnref:274" href="#fn:274"><sup>274</sup></a></li></ul> <h4>20th century</h4> <ul><li><a href="/facts/Lafforgue%27s_theorem/m2tULxrr">Lafforgue's theorem</a> (<a href="/facts/Laurent_Lafforgue/Q9cYLBB1">Laurent Lafforgue</a>, 1998)<a class="footnote-ref" id="fnref:275" href="#fn:275"><sup>275</sup></a></li> <li><a href="/facts/Fermat%27s_Last_Theorem/fHRc2t8z">Fermat's Last Theorem</a> (<a href="/facts/Andrew_Wiles/bdc3k3pQ">Andrew Wiles</a> and <a href="/facts/Richard_Taylor_(mathematician)/crBWfG0r">Richard Taylor</a>, 1995)<a class="footnote-ref" id="fnref:276" href="#fn:276"><sup>276</sup></a><a class="footnote-ref" id="fnref:277" href="#fn:277"><sup>277</sup></a></li></ul> <h3>Ramsey theory</h3> <ul><li><a href="/facts/Burr%E2%80%93Erd%C5%91s_conjecture/MEpNkTpz">Burr–Erdős conjecture</a> (Choongbum Lee, 2017)<a class="footnote-ref" id="fnref:278" href="#fn:278"><sup>278</sup></a></li> <li><a href="/facts/Boolean_Pythagorean_triples_problem/WllJtAtB">Boolean Pythagorean triples problem</a> (<a href="/facts/Marijn_Heule/rakzq0Ib">Marijn Heule</a>, Oliver Kullmann, <a href="/facts/Victor_W._Marek/bcFIqLTJ">Victor W. Marek</a>, 2016)<a class="footnote-ref" id="fnref:279" href="#fn:279"><sup>279</sup></a><a class="footnote-ref" id="fnref:280" href="#fn:280"><sup>280</sup></a></li></ul> <h3>Theoretical computer science</h3> <ul><li><a href="/facts/Decision_tree_model/FalJqMSd">Sensitivity conjecture</a> for Boolean functions (<a href="/facts/Hao_Huang_(mathematician)/vyakDDSq">Hao Huang</a>, 2019)<a class="footnote-ref" id="fnref:281" href="#fn:281"><sup>281</sup></a></li></ul> <h3>Topology</h3> <ul><li>Deciding whether the <a href="/facts/Conway_knot/qK4mk2Xo">Conway knot</a> is a <a href="/facts/Slice_knot/2IUXAXhc">slice knot</a> (<a href="/facts/Lisa_Piccirillo/ubFBWgn5">Lisa Piccirillo</a>, 2020)<a class="footnote-ref" id="fnref:282" href="#fn:282"><sup>282</sup></a><a class="footnote-ref" id="fnref:283" href="#fn:283"><sup>283</sup></a></li> <li><a href="/facts/Virtual_Haken_conjecture/4qgD4tWN">Virtual Haken conjecture</a> (<a href="/facts/Ian_Agol/ozxjQ99k">Ian Agol</a>, Daniel Groves, Jason Manning, 2012)<a class="footnote-ref" id="fnref:284" href="#fn:284"><sup>284</sup></a> (and by work of <a href="/facts/Daniel_Wise_(mathematician)/syYzYmEh">Daniel Wise</a> also <a href="/facts/Virtually_fibered_conjecture/1xRBcfTj">virtually fibered conjecture</a>)</li> <li><a href="/facts/Hsiang%E2%80%93Lawson%27s_conjecture/NUXcxSBM">Hsiang–Lawson's conjecture</a> (<a href="/facts/Simon_Brendle/kgEVRFCA">Simon Brendle</a>, 2012)<a class="footnote-ref" id="fnref:285" href="#fn:285"><sup>285</sup></a></li> <li><a href="/facts/Ehrenpreis_conjecture/J7o7yYBh">Ehrenpreis conjecture</a> (<a href="/facts/Jeremy_Kahn/UNnWwTAC">Jeremy Kahn</a>, <a href="/facts/Vladimir_Markovic/zQ5dHVPI">Vladimir Markovic</a>, 2011)<a class="footnote-ref" id="fnref:286" href="#fn:286"><sup>286</sup></a></li> <li><a href="/facts/Atiyah_conjecture/RCmEx56l">Atiyah conjecture</a> for groups with finite subgroups of unbounded order (Austin, 2009)<a class="footnote-ref" id="fnref:287" href="#fn:287"><sup>287</sup></a></li> <li><a href="/facts/Cobordism_hypothesis/ww5m31Tc">Cobordism hypothesis</a> (<a href="/facts/Jacob_Lurie/QUNLubLt">Jacob Lurie</a>, 2008)<a class="footnote-ref" id="fnref:288" href="#fn:288"><sup>288</sup></a></li> <li><a href="/facts/Spherical_space_form_conjecture/9T1FWPMg">Spherical space form conjecture</a> (<a href="/facts/Grigori_Perelman/SmISWW7x">Grigori Perelman</a>, 2006)</li> <li><a href="/facts/Poincar%C3%A9_conjecture/mxk2Pju3">Poincaré conjecture</a> (<a href="/facts/Grigori_Perelman/SmISWW7x">Grigori Perelman</a>, 2002)<a class="footnote-ref" id="fnref:289" href="#fn:289"><sup>289</sup></a></li> <li><a href="/facts/Geometrization_conjecture/FDpqIM7y">Geometrization conjecture</a>, (<a href="/facts/Grigori_Perelman/SmISWW7x">Grigori Perelman</a>,<a class="footnote-ref" id="fnref:290" href="#fn:290"><sup>290</sup></a> series of preprints in 2002–2003)<a class="footnote-ref" id="fnref:291" href="#fn:291"><sup>291</sup></a></li> <li><a href="/facts/Nikiel%27s_conjecture/nfKuuwYK">Nikiel's conjecture</a> (<a href="/facts/Mary_Ellen_Rudin/cYFs5kXH">Mary Ellen Rudin</a>, 1999)<a class="footnote-ref" id="fnref:292" href="#fn:292"><sup>292</sup></a></li> <li>Disproof of the <a href="/facts/Ganea_conjecture/8wErACra">Ganea conjecture</a> (Iwase, 1997)<a class="footnote-ref" id="fnref:293" href="#fn:293"><sup>293</sup></a></li></ul> <h3>Uncategorised</h3> <h4>2010s</h4> <ul><li><a href="/facts/Erd%C5%91s_discrepancy_problem/xmJ63B1X">Erdős discrepancy problem</a> (<a href="/facts/Terence_Tao/Do68JKn6">Terence Tao</a>, 2015)<a class="footnote-ref" id="fnref:294" href="#fn:294"><sup>294</sup></a></li> <li><a href="/facts/Umbral_moonshine/vBqoVR6p">Umbral moonshine</a> conjecture (John F. R. Duncan, Michael J. Griffin, <a href="/facts/Ken_Ono/Xw66ukAm">Ken Ono</a>, 2015)<a class="footnote-ref" id="fnref:295" href="#fn:295"><sup>295</sup></a></li> <li>Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (<a href="/facts/Jeff_Cheeger/VTmcprIR">Jeff Cheeger</a>, Aaron Naber, 2014)<a class="footnote-ref" id="fnref:296" href="#fn:296"><sup>296</sup></a></li> <li><a href="/facts/Gaussian_correlation_inequality/2WIbXcll">Gaussian correlation inequality</a> (<a href="/facts/Thomas_Royen/ey9fBA5d">Thomas Royen</a>, 2014)<a class="footnote-ref" id="fnref:297" href="#fn:297"><sup>297</sup></a></li> <li>Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, <a href="/facts/Aleksandar_Nikolov_(computer_scientist)/NAalPvVk">Aleksandar Nikolov</a>, 2011)<a class="footnote-ref" id="fnref:298" href="#fn:298"><sup>298</sup></a></li> <li><a href="/facts/Bloch%E2%80%93Kato_conjecture/0myRUsDm">Bloch–Kato conjecture</a> (<a href="/facts/Vladimir_Voevodsky/Fe7snF51">Vladimir Voevodsky</a>, 2011)<a class="footnote-ref" id="fnref:299" href="#fn:299"><sup>299</sup></a> (and <a href="/facts/Quillen%E2%80%93Lichtenbaum_conjecture/3JRUlNU1">Quillen–Lichtenbaum conjecture</a> and by work of Thomas Geisser and <a href="/facts/Marc_Levine_(mathematician)/l68TzYtr">Marc Levine</a> (2001) also <a href="/facts/Norm_residue_isomorphism_theorem/0myRUsDm">Beilinson–Lichtenbaum conjecture</a><a class="footnote-ref" id="fnref:300" href="#fn:300"><sup>300</sup></a><a class="footnote-ref" id="fnref:301" href="#fn:301"><sup>301</sup></a>: 359 <a class="footnote-ref" id="fnref:302" href="#fn:302"><sup>302</sup></a>)</li></ul> <h4>2000s</h4> <ul><li>Kauffman–Harary conjecture (Thomas Mattman, Pablo Solis, 2009)<a class="footnote-ref" id="fnref:303" href="#fn:303"><sup>303</sup></a></li> <li><a href="/facts/Surface_subgroup_conjecture/myeDSAVj">Surface subgroup conjecture</a> (<a href="/facts/Jeremy_Kahn/UNnWwTAC">Jeremy Kahn</a>, <a href="/facts/Vladimir_Markovic/zQ5dHVPI">Vladimir Markovic</a>, 2009)<a class="footnote-ref" id="fnref:304" href="#fn:304"><sup>304</sup></a></li> <li>Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Zhiqin Lu, 2007)<a class="footnote-ref" id="fnref:305" href="#fn:305"><sup>305</sup></a></li> <li>Nirenberg–Treves conjecture (<a href="/facts/Nils_Dencker/eCwThV57">Nils Dencker</a>, 2005)<a class="footnote-ref" id="fnref:306" href="#fn:306"><sup>306</sup></a><a class="footnote-ref" id="fnref:307" href="#fn:307"><sup>307</sup></a></li> <li><a href="/facts/Peter_Lax/sptzPICM">Lax conjecture</a> (<a href="/facts/Adrian_Lewis_(mathematician)/E8IUGK77">Adrian Lewis</a>, <a href="/facts/Pablo_Parrilo/9h8SuqY4">Pablo Parrilo</a>, Motakuri Ramana, 2005)<a class="footnote-ref" id="fnref:308" href="#fn:308"><sup>308</sup></a></li> <li>The <a href="/facts/Langlands%E2%80%93Shelstad_fundamental_lemma/cmEgFodJ">Langlands–Shelstad fundamental lemma</a> (<a href="/facts/Ng%C3%B4_B%E1%BA%A3o_Ch%C3%A2u/tIG28vye">Ngô Bảo Châu</a> and <a href="/facts/G%C3%A9rard_Laumon/Bta5Aa5B">Gérard Laumon</a>, 2004)<a class="footnote-ref" id="fnref:309" href="#fn:309"><sup>309</sup></a></li> <li><a href="/facts/Milnor_conjecture_(K-theory)/5vXHU1AM">Milnor conjecture</a> (<a href="/facts/Vladimir_Voevodsky/Fe7snF51">Vladimir Voevodsky</a>, 2003)<a class="footnote-ref" id="fnref:310" href="#fn:310"><sup>310</sup></a></li> <li>Kirillov's conjecture (Ehud Baruch, 2003)<a class="footnote-ref" id="fnref:311" href="#fn:311"><sup>311</sup></a></li> <li>Kouchnirenko's conjecture (Bertrand Haas, 2002)<a class="footnote-ref" id="fnref:312" href="#fn:312"><sup>312</sup></a></li> <li><a href="/facts/N!_conjecture/WIug3IIP"><i>n</i>! conjecture</a> (<a href="/facts/Mark_Haiman/U66xLVEH">Mark Haiman</a>, 2001)<a class="footnote-ref" id="fnref:313" href="#fn:313"><sup>313</sup></a> (and also <a href="/facts/Macdonald_polynomials/8gYD1p9g">Macdonald positivity conjecture</a>)</li> <li><a href="/facts/Kato%27s_conjecture/zyyXb7CL">Kato's conjecture</a> (<a href="/facts/Pascal_Auscher/eecXWsI9">Pascal Auscher</a>, <a href="/facts/Steve_Hofmann/fLGoz71Q">Steve Hofmann</a>, <a href="/facts/Michael_Lacey_(mathematician)/h0NBEUVT">Michael Lacey</a>, <a href="/facts/Alan_Gaius_Ramsay_McIntosh/enCXC7Mc">Alan McIntosh</a>, and Philipp Tchamitchian, 2001)<a class="footnote-ref" id="fnref:314" href="#fn:314"><sup>314</sup></a></li> <li>Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, <a href="/facts/Morihiko_Saito/Fv35Trr6">Morihiko Saito</a>, 2001)<a class="footnote-ref" id="fnref:315" href="#fn:315"><sup>315</sup></a></li> <li><a href="/facts/Modularity_theorem/aI3HRxwj">Modularity theorem</a> (<a href="/facts/Christophe_Breuil/IzljeQCV">Christophe Breuil</a>, <a href="/facts/Brian_Conrad/uR4ARcLT">Brian Conrad</a>, <a href="/facts/Fred_Diamond/fs32yCfw">Fred Diamond</a>, and <a href="/facts/Richard_Taylor_(mathematician)/crBWfG0r">Richard Taylor</a>, 2001)<a class="footnote-ref" id="fnref:316" href="#fn:316"><sup>316</sup></a></li> <li>Erdős–Stewart conjecture (<a href="/facts/Florian_Luca/YNGCrXKR">Florian Luca</a>, 2001)<a class="footnote-ref" id="fnref:317" href="#fn:317"><sup>317</sup></a></li> <li><a href="/facts/Berry%E2%80%93Robbins_problem/0SyrJkfd">Berry–Robbins problem</a> (<a href="/facts/Michael_Atiyah/RQVJ9ATE">Michael Atiyah</a>, 2000)<a class="footnote-ref" id="fnref:318" href="#fn:318"><sup>318</sup></a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/List_of_conjectures/vdrJkpYv">List of conjectures</a></li> <li><a href="/facts/List_of_unsolved_problems_in_statistics/FVU3YpcP">List of unsolved problems in statistics</a></li> <li><a href="/facts/List_of_unsolved_problems_in_computer_science/QySnBJ5E">List of unsolved problems in computer science</a></li> <li><a href="/facts/List_of_unsolved_problems_in_physics/vJ3d1A83">List of unsolved problems in physics</a></li> <li><a href="/facts/Lists_of_unsolved_problems/u9bpE2Xa">Lists of unsolved problems</a></li> <li><i><a href="/facts/Open_Problems_in_Mathematics/CPmWAp9Q">Open Problems in Mathematics</a></i></li> <li><i><a href="/facts/The_Great_Mathematical_Problems/gBXUb7mA">The Great Mathematical Problems</a></i></li> <li><a href="/facts/Scottish_Book/d4oQzY9s">Scottish Book</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="further-reading">Further reading</h2> <h3>Books discussing problems solved since 1995</h3> <ul><li><a href="/facts/Simon_Singh/fFstr4XL">Singh, Simon</a> (2002). <a href="/facts/Fermat%27s_Last_Theorem_(book)/RF45a1Yo"><i>Fermat's Last Theorem</i></a>. Fourth Estate. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-84115-791-7.</li> <li><a href="/facts/Donal_O%27Shea/3REwqWXK">O'Shea, Donal</a> (2007). <i>The Poincaré Conjecture</i>. Penguin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-84614-012-9.</li> <li><a href="/facts/George_Szpiro/7kSlMJYD">Szpiro, George G.</a> (2003). <i>Kepler's Conjecture</i>. Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-08601-7.</li> <li><a href="/facts/Mark_Ronan/ezsYQ0l2">Ronan, Mark</a> (2006). <i>Symmetry and the Monster</i>. Oxford. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-280722-9.</li></ul> <h3>Books discussing unsolved problems</h3> <ul><li><a href="/facts/Fan_Chung/amGEVrj5">Chung, Fan</a>; <a href="/facts/Ronald_Graham/ukKzczTL">Graham, Ron</a> (1999). <a href="/facts/Erd%C5%91s_on_Graphs/spjEfLIq"><i>Erdös on Graphs: His Legacy of Unsolved Problems</i></a>. AK Peters. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-111-6.</li> <li>Croft, Hallard T.; <a href="/facts/Kenneth_Falconer_(mathematician)/zf1e1BIG">Falconer, Kenneth J.</a>; <a href="/facts/Richard_K._Guy/shtfXH4N">Guy, Richard K.</a> (1994). <a href="https://archive.org/details/unsolvedproblems0000crof"><i>Unsolved Problems in Geometry</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-97506-1.</li> <li><a href="/facts/Richard_K._Guy/shtfXH4N">Guy, Richard K.</a> (2004). <i>Unsolved Problems in Number Theory</i>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-20860-2.</li> <li><a href="/facts/Victor_Klee/NMRFBGm6">Klee, Victor</a>; <a href="/facts/Stan_Wagon/7LGzJEN6">Wagon, Stan</a> (1996). <a href="https://archive.org/details/oldnewunsolvedpr0000klee"><i>Old and New Unsolved Problems in Plane Geometry and Number Theory</i></a>. The Mathematical Association of America. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-88385-315-3.</li> <li><a href="/facts/Marcus_du_Sautoy/7LabtSsz">du Sautoy, Marcus</a> (2003). <a href="https://archive.org/details/musicofprimes00marc"><i>The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics</i></a>. Harper Collins. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-06-093558-0.</li> <li><a href="/facts/John_Derbyshire/efG5wLFS">Derbyshire, John</a> (2003). <a href="https://archive.org/details/primeobsessionbe00derb_0"><i>Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics</i></a>. Joseph Henry Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-309-08549-6.</li> <li><a href="/facts/Keith_Devlin/9YsdjXjk">Devlin, Keith</a> (2006). <i>The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time</i>. Barnes & Noble. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7607-8659-8.</li> <li><a href="/facts/Vincent_Blondel/pSyIU7yU">Blondel, Vincent D.</a>; Megrestski, Alexandre (2004). <i>Unsolved problems in mathematical systems and control theory</i>. Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-11748-5.</li> <li><a href="/facts/Lizhen_Ji/PTHqKenj">Ji, Lizhen</a>; Poon, Yat-Sun; <a href="/facts/Shing-Tung_Yau/qIQclz7g">Yau, Shing-Tung</a> (2013). <i>Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics)</i>. International Press of Boston. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-57146-278-7.</li> <li><a href="/facts/Michel_Waldschmidt/LjyeFqxL">Waldschmidt, Michel</a> (2004). <a href="http://www.math.jussieu.fr/~miw/articles/pdf/odp.pdf">"Open Diophantine Problems"</a> (PDF). <i>Moscow Mathematical Journal</i>. 4 (1): 245–305. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0312440">math/0312440</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.17323%2F1609-4514-2004-4-1-245-305">10.17323/1609-4514-2004-4-1-245-305</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1609-3321">1609-3321</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:11845578">11845578</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1066.11030">1066.11030</a>.</li> <li><a href="/facts/Victor_Mazurov/dK1wGPC2">Mazurov, V. D.</a>; Khukhro, E. I. (1 Jun 2015). "Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1401.0300v6">1401.0300v6</a> [<a href="https://arxiv.org/archive/math.GR">math.GR</a>].</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://faculty.evansville.edu/ck6/integer/unsolved.html">24 Unsolved Problems and Rewards for them</a></li> <li><a href="http://www.openproblems.net/">List of links to unsolved problems in mathematics, prizes and research</a></li> <li><a href="http://garden.irmacs.sfu.ca/">Open Problem Garden</a></li> <li><a href="http://aimpl.org/">AIM Problem Lists</a></li> <li><a href="http://cage.ugent.be/~hvernaev/problems/archive.html">Unsolved Problem of the Week Archive</a>. MathPro Press.</li> <li><a href="/facts/John_M._Ball/ndeYYfV1">Ball, John M.</a> <a href="https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf">"Some Open Problems in Elasticity"</a> (PDF).</li> <li>Constantin, Peter. <a href="https://web.math.princeton.edu/~const/2k.pdf">"Some open problems and research directions in the mathematical study of fluid dynamics"</a> (PDF).</li> <li><a href="/facts/Denis_Serre/rBpUbeVq">Serre, Denis</a>. <a href="http://www.umpa.ens-lyon.fr/~serre/DPF/Ouverts.pdf">"Five Open Problems in Compressible Mathematical Fluid Dynamics"</a> (PDF).</li> <li><a href="http://unsolvedproblems.org/">Unsolved Problems in Number Theory, Logic and Cryptography</a></li> <li><a href="http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html">200 open problems in graph theory</a> <a href="https://web.archive.org/web/20170515145908/http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html">Archived</a> 2017-05-15 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li> <li><a href="http://cs.smith.edu/~orourke/TOPP/">The Open Problems Project (TOPP)</a>, discrete and computational geometry problems</li> <li><a href="http://math.berkeley.edu/~kirby/problems.ps.gz">Kirby's list of unsolved problems in low-dimensional topology</a></li> <li><a href="http://www.math.ucsd.edu/~erdosproblems/">Erdös' Problems on Graphs</a></li> <li><a href="https://arxiv.org/abs/1409.2823">Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory</a></li> <li><a href="http://www.sciencedirect.com/science/article/pii/S0165011414003194">Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications</a></li> <li><a href="http://wwwmath.uni-muenster.de/logik/Personen/rds/list.html">List of open problems in inner model theory</a></li> <li><a href="/facts/Michael_Aizenman/6TgAmny8">Aizenman, Michael</a>. <a href="https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/OPlist.html">"Open Problems in Mathematical Physics"</a>.</li> <li><a href="/facts/Barry_Simon/BImmcE2I">Barry Simon</a>'s <a href="http://math.caltech.edu/SimonPapers/R27.pdf">15 Problems in Mathematical Physics</a></li> <li><a href="/facts/Alexandre_Eremenko/ohfIII8t">Alexandre Eremenko</a>. <a href="https://www.math.purdue.edu/~eremenko/uns1.html">Unsolved problems in Function Theory</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Thiele, RĂŒdiger (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 MathĂ©matiques de la SMC. Vol. 21. pp. 243â295. ISBN 978-0-387-25284-1. <a href="978-0-387-25284-1" target="_blank">978-0-387-25284-1</a> <a href="#fnref:1" class="footnote-back-ref">â©</a></p></li> <li id="fn:2"><p>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.. <a href="978-1-4899-3585-4" target="_blank">978-1-4899-3585-4</a> <a href="#fnref:2" class="footnote-back-ref">â©</a></p></li> <li id="fn:3"><p>Shimura, G. (1989). "Yutaka Taniyama and his time". 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Retrieved 2018-07-07. <a href="https://web.archive.org/web/20180710010437/http://www2.cnrs.fr/en/2435.htm?debut=8&theme1=12" target="_blank">https://web.archive.org/web/20180710010437/http://www2.cnrs.fr/en/2435.htm?debut=8&theme1=12</a> <a href="#fnref:7" class="footnote-back-ref">â©</a></p></li> <li id="fn:8"><p>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. <a href="https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani" target="_blank">https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani</a> <a href="#fnref:8" class="footnote-back-ref">â©</a></p></li> <li id="fn:9"><p>Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. 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