数学上の未解決問題

その他の未解決問題

「―はいくつか」系

• 魔方陣の数はいくつあるか。
• 最小のシェルピンスキー数は 78557、最小のリーゼル数は 509203、最小のブリエ数は 3316923598096294713661 かどうか。
• シェルピンスキー数のうち、最小の素数は 271129 か。また、シェルピンスキー数の最初の2個は 78557、271129 か。
• 基底5における最小のシェルピンスキー数は 159986、最小のリーゼル数は 346802、最小のブリエ数は 120538009895207932[3] かどうか。
• 何個組までの社交数が存在するか。
• 3倍完全数は6個、4倍完全数は36個、5倍完全数は65個、6倍完全数は245個かどうか。
• ソファ問題 - L字型の通路を通すことができる、ソファの面積の最大値は何か。
• 接吻数問題
• レピュニットの問題
• グラハム問題

分野別

代数的数論

• Are there infinitely many real quadratic number fields with unique factorization (Class number problem)
• Characterize all algebraic number fields that have some power basis.
• Stark conjectures (including Brumer–Stark conjecture)

組合せ論

• Number of magic squares (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ですか？[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れですか？[18]
• Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net?[19]
• トムソンの問題英語版 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)?
• Pentagonal tiling - 15 types of convex pentagons are known to monohedrally tile the plane, and it is not known whether this list is complete.[20]
• 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
• 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]

出典

 [脚注の使い方]
1. ^ Helfgott, H.A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT]。
2. ^ Helfgott, H.A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT]。
3. ^ シェルピンスキー数としての被覆集合は{3, 13, 17, 313, 11489}、リーゼル数としての被覆集合は{3, 7, 19, 31, 829, 5167}である。
4. ^