数学上の未解決問題

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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)とは未だ解決されていない数学上の問題のことである。

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達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]

以下の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]

「―は無限に存在するか」系[編集]

「―は存在するか」系[編集]

  • 奇数の完全数は存在するか。
  • 準完全数は存在するか。
  • 2k の形をした数 (1, 2, 4, 8, 16, …) 以外に概完全数は存在するか。
  • 偶数と奇数の組の友愛数は存在するか。
  • 偶数同士、奇数同士の婚約数は存在するか。
  • 3つ組の社交数は存在するか。

「―は全て――」系[編集]

「―はいくつか」系[編集]

その他[編集]

分野別[編集]

加法的整数論[編集]

{{:en:seealso|Problems involving arithmetic progressions}}

代数[編集]

代数幾何[編集]

  • André–Oort conjecture
  • Bass conjecture
  • Deligne conjecture
  • Fröberg conjecture
  • Fujita conjecture
  • Hartshorne conjectures
  • Manin conjecture
  • Nakai conjecture
  • Resolution of singularities in characteristic p
  • Standard conjectures on algebraic cycles
  • Section conjecture
  • Tate conjecture
  • Virasoro conjecture
  • Whitehead conjecture
  • Zariski multiplicity conjecture

代数的数論[編集]

  • 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)

解析[編集]

  • The Jacobian conjecture
  • Schanuel's conjecture and four exponentials conjecture
  • Lehmer's conjecture
  • Pompeiu problem
  • Are (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, πe, π√2, ππ, eπ2, ln π, 2e, ee, Catalan's constant or Khinchin's constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?[11][12][13]
  • Khabibullin’s conjecture on integral inequalities
  • Hilbert's thirteenth problem
  • Vitushkin's conjecture

組合せ論[編集]

  • Number of magic squares (sequence A006052 in OEIS)
  • Number of magic tori (sequence A270876 in OEIS)
  • Finding a formula for the probability that two elements chosen at random generate the symmetric group
  • Frankl's 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
  • The 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?
  • Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
  • The 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]

  • 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
  • Kakeya conjecture

ユークリッド幾何学[編集]

  • 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 – what is the smallest area of a shape that can cover every unit-length curve in the plane?[17]
  • The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?[18]
  • Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net?[19]
  • 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[編集]

  • 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]

  • 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
  • Hadwiger conjecture (graph theory)
  • 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[編集]

  • 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?

モデル理論[編集]

  • 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]

  • Grand Riemann hypothesis
    • Generalized Riemann hypothesis
      • Riemann hypothesis
  • n conjecture
    • abc conjecture (Proof claimed in 2012, currently under review.)
  • Hilbert's ninth problem
  • Hilbert's eleventh problem
  • Hilbert's twelfth problem
  • Carmichael's totient function conjecture
  • Erdős–Straus conjecture
  • Pillai's conjecture
  • Hall's conjecture
  • Lindelöf hypothesis
  • Montgomery's pair correlation conjecture
  • Hilbert–Pólya conjecture
  • Grimm's conjecture
  • Leopoldt's conjecture
  • Do any odd perfect numbers exist?
  • Are there infinitely many perfect numbers?
  • Do quasiperfect numbers exist?
  • Do any odd weird numbers exist?
  • Do any Lychrel numbers exist?
  • Is 10 a solitary number?
  • Catalan–Dickson conjecture on aliquot sequences
  • Do any Taxicab(5, 2, n) exist for n>1?
  • Brocard's problem: existence of integers, (n,m), such that n!+1 = m2 other than n=4, 5, 7
  • Beilinson conjecture
  • Littlewood conjecture
  • Szpiro's conjecture
  • Vojta's conjecture
  • 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?
  • Are there infinitely many amicable numbers?
  • Are there any pairs of amicable numbers which have opposite parity?
  • Are there any pairs of relatively prime amicable numbers?
  • Are there infinitely many betrothed numbers?
  • 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
  • Sato–Tate conjecture
  • Find value of De Bruijn–Newman constant
  • Which integers can be written as the sum of three perfect cubes?

近年解かれた問題[編集]


出典[編集]

  1. ^ Helfgott, H.A. (2013年). “Major arcs for Goldbach's theorem”. arXiv:1305.2897 [math.NT]. 
  2. ^ Helfgott, H.A. (2012年). “Minor arcs for Goldbach's problem”. arXiv:1205.5252/ [math.NT]. 
  3. ^ “数学の難問「ABC予想」京大教授が証明成功か”. 読売新聞. (2012年9月19日). http://www.yomiuri.co.jp/science/news/20120919-OYT1T00490.htm 2012年9月19日閲覧。 
  4. ^ 加藤敏夫カリフォルニア大学バークレー校教授。

関連項目[編集]