数学上の未解決問題(すうがくじょうのみかいけつもんだい、英: unsolved problems in mathematics)とは、未だ解決されていない数学上の問題のことで、未解決問題の定義を「未だ証明が得られていない命題」という立場を取るのであれば、そういった問題は数学界に果てしなく存在する。ここでは、リーマン予想のようにその証明結果が数学全域と関わりを持つような命題、P≠NP予想のようにその結論が現代科学、技術のあり方に甚大な影響を及ぼす可能性があるような命題、問いかけのシンプルさ故に数多くの数学者や数学愛好家たちが証明を試みてきたような有名な命題を列挙する。
ウェアリングの問題 (The values of g(k) and G(k) in en:Waring's problem)
コラッツ予想 (en:Collatz conjecture)(3n + 1 conjecture)
ランダー・パーキン・セルフリッジ予想
Diophantine quintuples
ギルブレース予想
等差数列に関するエルデシュ予想
Erdős–Turán conjecture on additive bases
Pollock octahedral numbers conjecture
スコーレム問題
Determine growth rate of rk(N) (see Szemerédi's theorem)
Minimum overlap problem
代数
可換環のホモロジカル予想
ヒルベルトの23の問題の16番目 (en:Hilbert's sixteenth problem)
ヒルベルトの23の問題の15番目 (en:Hilbert's fifteenth problem)
アダマール予想 (en:Hadamard conjecture)
ジャコブソン予想 (en:Jacobson's conjecture)
Existence of (perfect cuboids) and associated (en:Cuboid conjectures)
(Zauner's conjecture): existence of (en:SIC-POVM)s in all dimensions
(Wild Problem): Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
ケーテ予想 (en:Köthe conjecture)
(en:Birch–Tate conjecture)
セール予想 (Serre's conjecture II)
(en:Bombieri–Lang conjecture)
(en:Farrell–Jones conjecture)
ボスト予想 (Bost conjecture)
(en:Uniformity conjecture)
(en:Kaplansky's conjecture)
(en:Kummer–Vandiver conjecture)
(en:Serre's multiplicity conjectures)
(en:Pierce–Birkhoff conjecture)
(en:Eilenberg–Ganea conjecture)
グリーン予想
(en:Grothendieck–Katz p-curvature conjecture)
(en:Sendov's conjecture)
ゴールマハティヒ予想 (en:Goormaghtigh conjecture)
代数幾何
アンドレ・オールト予想(André–Oort conjecture)
バスの予想(Bass conjecture)
Deligne conjecture
Fröberg conjecture
藤田予想
Hartshorne conjectures
Manin conjecture
中井予想
Resolution of singularities in characteristic p
代数的サイクルの標準予想(en:Standard conjectures on algebraic cycles)
Section conjecture
テイト予想 (代数幾何学)(en:Tate conjecture)
Virasoro conjecture
Whitehead conjecture
Zariski multiplicity conjecture
代数的数論
Are there infinitely many real quadratic number fields with unique factorization (類数問題)
Characterize all algebraic number fields that have some power basis.
Stark conjectures (including Brumer–Stark conjecture)
解析
ヤコビアン予想(The Jacobian conjecture)
Schanuel's conjecture and four exponentials conjecture
レーマーの予想(en:Lehmer's conjecture)
Pompeiu problem
γ(オイラーの定数),π + e, π − e, πe, π/e, πe, π√2, ππ、 eπ2, ln π, 2e, ee, カタランの定数, ヒンチンの定数, これらは代数的無理数(代数的数であり、無理数でもある数)か、超越数か?これらの無理数度の値は何か?
Khabibullin’s conjecture on integral inequalities
ヒルベルトの23の問題の13番目(en:Hilbert's thirteenth problem)
Vitushkin's conjecture
組合せ論
魔方陣の数 (sequence A006052 in OEIS [1])
Number of magic tori (sequence A270876 in OEIS [2])
Finding a formula for the probability that two elements chosen at random generate the symmetric group
en:Union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
en:Lonely runner conjecture: if runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance from each other runner) at some time?
en:Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
en:1/3–2/3 conjecture : does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?
unicity conjecture for Markov numbers
balance puzzle [14]
離散幾何学
Solving the happy ending problem for arbitrary
Finding matching upper and lower bounds for k-sets and halving lines
The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies
The Kobon triangle problem on triangles in line arrangements
The McMullen problem on projectively transforming sets of points into convex position
Ulam's packing conjecture about the identity of the worst-packing convex solid
Filling area conjecture
Hopf conjecture
掛谷予想(Kakeya conjecture)
ユークリッド幾何学
The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?[15]
Inscribed square problem – does every Jordan curve have an inscribed square?[16]
Moser's worm problem – 平面内のすべての単位長曲線をカバーできる形状の最小領域は何か?[17]
ソファ問題 – 単位幅のL字型の廊下を通過できる形状の最大領域はどんな形か?[18]
Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net?[19]
トムソン問題 The Thomson problem - what is the minimum energy configuration of N particles bound to the surface of a unit sphere that repel each other with a 1/r potential (or any potential in general)?
Falconer's conjecture
g-conjecture
Circle packing in an equilateral triangle
Circle packing in an isosceles right triangle
力学系
Furstenberg conjecture – Is every invariant and ergodic measure for the action on the circle either Lebesgue or atomic?
Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups
MLC conjecture – Is the Mandelbrot set locally connected ?
Weinstein conjecture - Does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?
Is every reversible cellular automaton in three or more dimensions locally reversible?[21]
グラフ理論
Barnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle
The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
The Erdős–Hajnal conjecture on finding large homogeneous sets in graphs with a forbidden induced subgraph
The Hadwiger conjecture relating coloring to clique minors
The Erdős–Faber–Lovász conjecture on coloring unions of cliques
Harborth's conjecture that every planar graph can be drawn with integer edge lengths
The total coloring conjecture
The list coloring conjecture
Hadwiger conjecture (en:Hadwiger conjecture)
The Ringel–Kotzig conjecture on graceful labeling of trees
How many unit distances can be determined by a set of n points? (see Counting unit distances)
The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
Lovász conjecture
Deriving a closed-form expression for the percolation threshold values, especially (square site)
Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow
Petersen coloring conjecture
The reconstruction conjecture and new digraph reconstruction conjecture concerning whether or not a graph is recognizable by the vertex deleted subgraphs.
The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice.
Does a Moore graph with girth 5 and degree 57 exist?
Conway's thrackle conjecture
Negami's conjecture on the characterization of graphs with planar covers
The Blankenship–Oporowski conjecture on the book thickness of subdivisions
Hedetniemi's conjecture
Vizing's conjecture
群論
Is every finitely presented periodic group finite?
The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
Is every group surjunctive?
Andrews–Curtis conjecture
Herzog–Schönheim conjecture
Does generalized moonshine exist?
コクセター群の同型問題
モデル理論
Vaught's conjecture
The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in is a simple algebraic group over an algebraically closed field.
The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for -saturated models of a countable theory.[22]
Determine the structure of Keisler's order[23][24]
The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
Is the theory of the field of Laurent series over decidable? of the field of polynomials over ?
(BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[25]
The Stable Forking Conjecture for simple theories[26]
For which number fields does Hilbert's tenth problem hold?
Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality does it have a model of cardinality continuum?[27]
Shelah's eventual Categority conjecture: For every cardinal \lambda there exists a cardinal \mu(\lambda) such that If an AEC K with LS(K)<= \lambda is categorical in a cardinal above \mu(\lambda) then it is categorical in all cardinals above \mu(\lambda).[22][28]
Shelah's categoricity conjecture for L_{\omega_1,\omega}: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[22]
Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[29]
If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?[30][31]
Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
Kueker's conjecture[32]
Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
Lachlan's decision problem
Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?[33]
The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[34]
数論
大リーマン予想(Grand Riemann hypothesis)
一般化されたリーマン予想(en:Generalized Riemann hypothesis)
リーマン予想(en:Riemann hypothesis)
n conjecture
ヒルベルトの23の問題の9番目(en:Hilbert's ninth problem)
ヒルベルトの23の問題の11番目(en:Hilbert's eleventh problem)
ヒルベルトの23の問題の第12の問題 (en:Hilbert's twelfth problem)
Carmichael's totient function conjecture
Erdős–Straus conjecture
Pillai's conjecture
マーシャル・ホール予想
Lindelöf hypothesis
Montgomery's pair correlation conjecture
Hilbert–Pólya conjecture
Grimm's conjecture
Leopoldt's conjecture
Do any odd perfect numbers exist?
完全数は、無限にあるか。
準完全数は存在するか。
Do any odd weird numbers exist?
リクレル数は存在するか。
Is 10 a solitary number?
Catalan–Dickson conjecture on aliquot sequences
Do any Taxicab(5, 2, n) exist for n>1?
ブロカールの問題(Brocard's problem: existence of integers, (n,m), such that n!+1 = m2 other than n=4, 5, 7)
Beilinson conjecture
Littlewood conjecture
スピロ予想(en:Szpiro's conjecture)
ヴォイタ予想(en:Vojta's conjecture)
ゴールマハティヒ予想(en:Goormaghtigh conjecture)
Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
Lehmer's totient problem: if φ(n) divides n − 1, must n be prime?
友愛数は無数にあるか?
Are there any pairs of amicable numbers which have opposite parity?
互いに素な友愛数のペアはあるか?
婚約数は無数にあるか?
Are there any pairs of betrothed numbers which have same parity?
The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?
Is π a normal number (its digits are "random")?[35]
Casas-Alvero conjecture
Find value of De Bruijn–Newman constant
3つの立方数の和 : Which integers can be written as the sum of three perfect cubes?