数学上の未解決問題

出典: フリー百科事典『ウィキペディア（Wikipedia）』

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。

ミレニアム懸賞問題[編集]

詳細は「ミレニアム懸賞問題」を参照

以下7つの問題はミレニアム懸賞問題と呼ばれ、クレイ数学研究所によってそれぞれ100万ドルの懸賞金が懸けられている。

その他の未解決問題[編集]

「―は無数に存在するか」系[編集]

「―は存在するか（否か）」系[編集]

「―は全て――」系[編集]

「―はいくつか」系[編集]

  • 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[1] 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。
  • 素数計数関数 π (x) が対数積分 li (x) より大きくなる最小の x は何か。
  • 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

その他[編集]

分野別[編集]

加法的整数論[編集]

加法的整数論については、加法的整数論（英語版）を参照されたい。

代数[編集]

代数幾何については「代数幾何」を参照

代数幾何[編集]

代数幾何については「代数幾何」を参照

代数的数論[編集]

代数的数論については「代数的数論」を参照
  • 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[編集]

解析については「解析」を参照

組合せ論[編集]

組合せ論については「組合せ論」を参照
  • 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[編集]

代数幾何については「代数幾何」を参照
  • 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[編集]

ユークリッド幾何学については「ユークリッド幾何学」を参照
  • 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[編集]

力学系については「力学系」を参照
  • 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[編集]

グラフ理論については「グラフ理論」を参照
  • 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（英語版）

群論[編集]

群論については「群論」を参照
  • 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[編集]

モデル理論については「モデル理論」を参照
  • 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[編集]

数論については「数論」を参照

近年解かれた問題[編集]

出典[編集]

[脚注の使い方]
  1. ^ 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。
  2. ^ Wolchover, Natalie (July 11, 2017), “Pentagon Tiling Proof Solves Century-Old Math Problem”, Quanta Magazine, オリジナルのAugust 6, 2017時点におけるアーカイブ。, https://web.archive.org/web/20170806093353/https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/ 2017年7月18日閲覧。 
  3. ^ Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT]。
  4. ^ Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT]。
  5. ^ Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT]。

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