List of theorems

This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures List of derivatives and integrals in alternative calculi List of equations List of fundamental theorems List of hypotheses List of inequalities Lists of integrals List of laws List of lemmas List of limits List of logarithmic identities List of mathematical functions List of mathematical identities List of mathematical proofs List of misnamed theorems List of scientific laws List of theories

Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields.

Logics and foundations

Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical logic) Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory) Church–Rosser theorem (lambda calculus) Compactness theorem (mathematical logic) Conservativity theorem (mathematical logic) Craig's theorem (mathematical logic) Craig's interpolation theorem (mathematical logic) Cut-elimination theorem (proof theory) Deduction theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set theory) Feferman–Vaught theorem (model theory) Friedberg–Muchnik theorem (mathematical logic) Fundamental theorem of equivalence relations (set theory) Glivenko's theorem (mathematical logic) Gentzen's consistency proof (proof theory) Gödel's completeness theorem (mathematical logic) Gödel's incompleteness theorem (mathematical logic) Goodstein's theorem (mathematical logic) Herbrand's theorem (logic) Independence of the axiom of choice (mathematical logic) Independence of the continuum hypothesis (mathematical logic) Kanamori–McAloon theorem (mathematical logic) Kirby–Paris theorem (proof theory) Kleene's recursion theorem (recursion theory) König's theorem (set theory, mathematical logic) Lindström's theorem (mathematical logic) Löb's theorem (mathematical logic) Łoś' theorem (model theory) Löwenheim–Skolem theorem (mathematical logic) Matiyasevich's theorem (mathematical logic) Morley's categoricity theorem (model theory) Paris–Harrington theorem (mathematical logic) Post's theorem (mathematical logic) Richardson's theorem (mathematical logic) Robinson's joint consistency theorem (mathematical logic) Sahlqvist correspondence theorem (modal logic) Soundness theorem (mathematical logic) Tarski's indefinability theorem (mathematical logic) Tennenbaum's theorem (model theory) Uncountability of the continuum (set theory) Unique homomorphic extension theorem (mathematical logic) Well-ordering theorem (mathematical logic) Wilkie's theorem (model theory) Zorn's lemma (set theory)

Combinatorics

2-factor theorem (graph theory) Abel's binomial theorem (combinatorics) Alspach's theorem (graph theory) Aztec diamond theorem (combinatorics) BEST theorem (graph theory) Baranyai's theorem (combinatorics) Berge's theorem (graph theory) Binomial theorem (algebra, combinatorics) Bondy's theorem (graph theory, combinatorics) Bondy–Chvátal theorem (graph theory) Brooks's theorem (graph theory) Bruck–Chowla–Ryser theorem (combinatorics) Cameron–Erdős theorem (discrete mathematics) Corners theorem (arithmetic combinatorics) Courcelle's theorem (graph theory) De Bruijn–Erdős theorem (graph theory) Dirac's theorems (graph theory) Erdős–Gallai theorem (graph theory) Erdős–Ginzburg–Ziv theorem (number theory) Erdős–Ko–Rado theorem (combinatorics) Erdős–Pósa theorem (graph theory) Erdős–Stone theorem (graph theory) Euler's partition theorem (number theory) Fermat polygonal number theorem (number theory) Five color theorem (graph theory) Four color theorem (graph theory) Freiman's theorem (number theory) Friendship theorem (graph theory) Galvin's theorem (combinatorics) Gomory's theorem (combinatorics) Graph structure theorem (graph theory) Grinberg's theorem (graph theory) Grötzsch's theorem (graph theory) Hajnal–Szemerédi theorem (graph theory) Hales–Jewett theorem (combinatorics) Hall's marriage theorem (combinatorics) Halpern–Läuchli theorem (Ramsey theory) Hindman's theorem (Ramsey theory) Kirchhoff's theorem (graph theory) Kneser's theorem (combinatorics) Kőnig's theorem (bipartite graphs) Kövari–Sós–Turán theorem (graph theory) Kruskal–Katona theorem (combinatorics) Kuratowski's theorem (graph theory) Lambek–Moser theorem (combinatorics) MacMahon's master theorem (enumerative combinatorics) Menger's theorem (graph theory) Milliken–Taylor theorem (Ramsey theory) Milliken's tree theorem (Ramsey theory) Multinomial theorem (algebra, combinatorics) Mycielski's theorem (graph theory) Nicomachus's theorem (number theory) Ore's theorem (graph theory) Paley's theorem (algebra) Perfect graph theorem (graph theory) Perlis theorem (graph theory) Planar separator theorem (graph theory) Pólya enumeration theorem (combinatorics) Ramsey's theorem (graph theory, combinatorics) Ringel–Youngs theorem (graph theory) Robbins' theorem (graph theory) Robertson–Seymour theorem (graph theory) Schnyder's theorem (graph theory) Schur's theorem (Ramsey theory) Schwenk's theorem (graph theory) Sensitivity theorem (computational complexity theory) Sperner's theorem (combinatorics) Stanley's reciprocity theorem (combinatorics) Star of David theorem (combinatorics) Stirling's theorem (mathematical analysis) Strong perfect graph theorem (graph theory) Symmetric hypergraph theorem (graph theory) Szemerédi's theorem (combinatorics) Theorem on friends and strangers (Ramsey theory) Tutte's theorem on perfect matchings (graph theory) Turán's theorem (graph theory) Van der Waerden's theorem (combinatorics) Vizing's theorem (graph theory) Wagner's theorem (graph theory) Zeilberger–Bressoud theorem (combinatorics)

Order, lattices, ordered algebraic structures

Birkhoff's representation theorem (lattice theory) Boolean prime ideal theorem (mathematical logic) Bourbaki–Witt theorem (order theory) Cantor's isomorphism theorem (order theory) Dilworth's theorem (combinatorics, order theory) Four functions theorem (combinatorics) Hahn embedding theorem (ordered groups) Hausdorff maximality theorem (set theory) Kleene fixed-point theorem (order theory) Knaster–Tarski theorem (order theory) Kruskal's tree theorem (order theory) Shannon's expansion theorem (Boolean algebra) Stone's representation theorem for Boolean algebras (mathematical logic) Szpilrajn extension theorem (axiom of choice)

General algebraic systems

Fundamental theorem on homomorphisms (abstract algebra) Isomorphism theorem (abstract algebra) Lattice theorem (abstract algebra)

Number theory

15 and 290 theorems (number theory) Albert–Brauer–Hasse–Noether theorem (algebras) Ankeny–Artin–Chowla theorem (number theory) Apéry's theorem (number theory) Artin–Verdier duality theorem (number theory) ATS theorem (number theory) Auxiliary polynomial theorem (Diophantine approximation) Ax–Kochen theorem (number theory) Baker's theorem (number theory) Barban–Davenport–Halberstam theorem (analytic number theory) Basel problem (mathematical analysis) Beatty's theorem (Diophantine approximation) Behrend's theorem (number theory) Bertrand's postulate (number theory) Birch's theorem (algebraic number theory) Bombieri's theorem (number theory) Bombieri–Friedlander–Iwaniec theorem (number theory) Brauer–Siegel theorem (number theory) Brun's theorem (number theory) Brun–Titchmarsh theorem (number theory) Carmichael's theorem (Fibonacci numbers) Chebotarev's density theorem (number theory) Chen's theorem (number theory) Chowla–Mordell theorem (number theory) Cohn's irreducibility criterion (polynomials) Critical line theorem (number theory) Davenport–Schmidt theorem (number theory, Diophantine approximations) Dirichlet's approximation theorem (Diophantine approximations) Dirichlet's theorem on arithmetic progressions (number theory) Dirichlet's unit theorem (algebraic number theory) Equidistribution theorem (ergodic theory) Erdős–Kac theorem (number theory) Euclid's theorem (number theory) Euclid–Euler theorem (number theory) Euler's theorem (number theory) Fermat's Last Theorem (number theory) Fermat's little theorem (number theory) Fermat's theorem on sums of two squares (number theory) Ferrero–Washington theorem (algebraic number theory) Ford's theorem (number theory) Franel–Landau theorem (number theory) Gelfond–Schneider theorem (transcendental number theory) Glaisher's theorem (number theory) Green–Tao theorem (number theory) Gross–Zagier theorem (number theory) Grunwald–Wang theorem (algebraic number theory) Hardy–Ramanujan theorem (number theory) Hasse norm theorem (number theory) Hasse–Minkowski theorem (number theory) Herbrand–Ribet theorem (cyclotomic fields) Hilbert–Speiser theorem (cyclotomic fields) Hilbert–Waring theorem (number theory) Hilbert's irreducibility theorem (number theory) Hurwitz's theorem (number theory) Jacobi's four-square theorem (number theory) Jurkat–Richert theorem (analytic number theory) Kaplansky's theorem on quadratic forms (number theory) Khinchin's theorem (probability) Kronecker's theorem (Diophantine approximation) Kronecker–Weber theorem (number theory) Kummer's theorem (number theory) Lafforgue's theorem (algebraic number theory) Lagrange's four-square theorem (number theory) Landau prime ideal theorem (number theory) Lindemann–Weierstrass theorem (transcendental number theory) Linnik's theorem (number theory) Lochs's theorem (number theory) Lucas's theorem (number theory) Mahler's compactness theorem (geometry of numbers) Mahler's theorem (p-adic analysis) Maier's theorem (analytic number theory) Mann's theorem (number theory) Mazur's control theorem (number theory) Mertens's theorems (number theory) Midy's theorem (number theory) Mihăilescu's theorem (number theory) Mirsky–Newman theorem (group theory) Modularity theorem (number theory) Mordell–Weil theorem (number theory) Multiplicity-one theorem (group representations) Nagell–Lutz theorem (elliptic curves) Niven's theorem (number theory) Ostrowski's theorem (number theory) Pentagonal number theorem (number theory) Prime number theorem (number theory) Principal ideal theorem (algebraic number theory) Proth's theorem (number theory) Quadratic reciprocity theorem Ramanujan–Skolem's theorem (Diophantine equations) Reflection theorem (algebraic number theory) Ribet's theorem (elliptic curves) Robin's theorem (number theory) Rosser's theorem (number theory) Siegel–Walfisz theorem (analytic number theory) Six exponentials theorem (transcendental number theory) Skolem–Mahler–Lech theorem (number theory) Solutions to Pell's equation (number theory) Sophie Germain's theorem (number theory) Sphere packing theorems in dimensions 8 and 24 (geometry, modular forms) Stark–Heegner theorem (number theory) Subspace theorem (Diophantine approximation) Sylvester's theorem (number theory) Takagi existence theorem (number theory) Thabit ibn Qurra's theorem (amicable numbers) Thue's theorem (Diophantine equation) Thue–Siegel–Roth theorem (Diophantine approximation) Tijdeman's theorem (Diophantine equations) Tunnell's theorem (number theory) Turán–Kubilius theorem (number theory) Vantieghems theorem (number theory) Vinogradov's theorem (number theory) Von Staudt–Clausen theorem (number theory) Wilson's theorem (number theory) Wolstenholme's theorem (number theory) Zeckendorf's theorem (number theory) Zsigmondy's theorem (number theory)

Field theory and polynomials

Abel–Ruffini theorem (theory of equations, Galois theory) Artin–Schreier theorem (real closed fields) Chevalley–Warning theorem (field theory) Diller–Dress theorem (field theory) Fundamental theorem of Galois theory (Galois theory) Hasse–Arf theorem (local class field theory) Hilbert's theorem 90 (number theory)