List of mathematical proofs - AbsoluteAstronomy.com

Theorems of which articles are primarily devoted to proving them

See also: :Category:Article proofs

Bertrand's postulate and a proof
Estimation of covariance matrices
Estimation of covariance matrices
In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution...
Fermat's little theorem
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...

and some proofs
Proofs of Fermat's little theorem
This article collects together a variety of proofs of Fermat's little theorem, which states thata^p \equiv a \pmod p \,\!for every prime number p and every integer a .-Simplifications:...
Gödel's completeness theorem
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....

and its original proof
Original proof of Gödel's completeness theorem
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 is not easy to read today; it uses concepts and formalism that are outdated and terminology that is often obscure...
Mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

and a proof
Proof that 0.999... equals 1
Proof that 22/7 exceeds π
Proof that e is irrational
Proof that e is irrational
In mathematics, the series representation of Euler's number ecan be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function evaluated at y = 1.-Summary of the proof:...
Proof that the sum of the reciprocals of the primes diverges

Articles devoted to theorems of which a (sketch of a) proof is given

See also: :Category:Articles containing proofs

Banach fixed point theorem
Banach fixed point theorem
In mathematics, the Banach fixed-point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points...
Banach–Tarski paradox
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...
Bolzano-Weierstrass theorem
Brouwer fixed point theorem
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
Buckingham π theorem (proof in progress)
Burnside's lemma
Burnside's lemma
Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George...
Cantor's theorem
Cantor's theorem
In elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
Cantor–Bernstein–Schroeder theorem
Cantor–Bernstein–Schroeder theorem
In set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions and between the sets A and B, then there exists a bijective function...
Cayley's formula
Cayley's theorem
Cayley's theorem
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
Clique problem
Clique problem
In computer science, the clique problem refers to any of the problems related to finding particular complete subgraphs in a graph, i.e., sets of elements where each pair of elements is connected....

(to do)
Compactness theorem
Compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model...

(very compact proof)
Erdős-Ko-Rado theorem
Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
Euler's four-square identity
Euler's four-square identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which being a sum of four squares, is itself a sum of four squares. Specifically:=\,...
Euler's theorem
Five color theorem
Five color theorem
The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than five colors in such a way that no two adjacent regions receive the same color.The five color theorem is...
Five lemma
Five lemma
In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams....
Fundamental theorem of arithmetic
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
Gauss-Markov theorem (brief pointer to proof)
Gödel's incompleteness theorem
- Gödel's first incompleteness theorem
- Gödel's second incompleteness theorem
Goodstein's theorem
Goodstein's theorem
In mathematical logic, Goodstein's theorem is a statement about the natural numbers, made by Reuben Goodstein, which states that every Goodstein sequence eventually terminates at 0. showed that it is unprovable in Peano arithmetic...
Green's theorem
Green's theorem
In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...

(to do)
- Green's theorem when D is a simple region
Heine-Borel theorem
Intermediate value theorem
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
Itō's lemma
Ito's lemma
In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...
König's lemma
König's lemma
König's lemma or König's infinity lemma is a theorem in graph theory due to Dénes Kőnig . It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic,...
König's theorem (set theory)
König's theorem (set theory)
In set theory, König's theorem colloquially states that if the axiom of choice holds, I is a set, mi and ni are cardinal numbers for every i in I, and m_i In set theory, König's theorem In set theory, König's theorem (named after the Hungarian mathematician Gyula Kőnig, who published under the...
König's theorem (graph theory)
König's theorem (graph theory)
In the mathematical area of graph theory, König's theorem, proved by Dénes Kőnig in 1931, describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs...
Lagrange's theorem
Lagrange's theorem
In mathematics, Lagrange's theorem usually refers to any of the following theorems, attributed to Joseph Louis Lagrange:* Lagrange's theorem * Lagrange's theorem...
Liouville's theorem
Liouville's theorem
Liouville's theorem has various meanings, all mathematical results named after Joseph Liouville:*In complex analysis, see Liouville's theorem ; there is also a related theorem on harmonic functions....

(brief pointer to proof)
Markov's inequality
Markov's inequality
In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant...

(proof of a generalization)
Mean value theorem
Mean value theorem
In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
Multivariate normal distribution (to do)
Holomorphic functions are analytic
Holomorphic functions are analytic
In complex analysis, a branch of mathematics, a complex-valued function ƒ of a complex variable z.*is said to be holomorphic at a point a if it is differentiable at every point within some open disk centered at a, and...
Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
Quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
Quotient rule
Ramsey's theorem
Ramsey's theorem
In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs...
Rao-Blackwell theorem
Rice's theorem
Rice's theorem
In computability theory, Rice's theorem states that, for any non-trivial property of partial functions, there is no general and effective method to decide whether an algorithm computes a partial function with that property...
Rolle's theorem
Rolle's theorem
In calculus, Rolle's theorem essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero.-Standard version of the theorem:If a real-valued function ƒ is continuous on a closed...
Splitting lemma
Splitting lemma
In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent....
squeeze theorem
Squeeze theorem
In calculus, the squeeze theorem is a theorem regarding the limit of a function....
Sum rule in differentiation
Sum rule in differentiation
In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation. The sum rule in integration follows from it...
Sum rule in integration
Sum rule in integration
In calculus, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. It is of particular use for the integration of sums, and is one part of the linearity of integration....
Sylow theorems
Transcendence of e and π (as corollaries of Lindemann-Weierstrass)
Tychonoff's theorem
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...

(to do)
Ultrafilter lemma
Ultraparallel theorem
Ultraparallel theorem
In hyperbolic geometry, the ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular hyperbolic line.-Proof in the Poincaré half-plane model:Leta...
Urysohn's lemma
Urysohn's lemma
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function....
Van der Waerden's theorem
Wilson's theorem
Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...

Articles devoted to algorithms in which their correctness is proven

Bellman-Ford algorithm
Bellman-Ford algorithm
The Bellman–Ford algorithm computes single-source shortest paths in a weighted digraph.For graphs with only non-negative edge weights, the faster Dijkstra's algorithm also solves the problem....

(to do)
Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
Kruskal's algorithm
Kruskal's algorithm
Kruskal's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized...
Gale–Shapley algorithm
Prim's algorithm
Prim's algorithm
In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a connected weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized...
Shor's algorithm
Shor's algorithm
Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm for integer factorization formulated in 1994...

(incomplete)

Articles where example statements are proven

See also: :Category:Articles containing proofs

Basis (linear algebra)
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
Burrows-Abadi-Needham logic
Burrows-Abadi-Needham logic
Burrows–Abadi–Needham logic is a set of rules for defining and analyzing information exchange protocols. Specifically, BAN logic helps its users determine whether exchanged information is trustworthy, secured against eavesdropping, or both...
Direct proof
Direct proof
In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions...
Generating a vector space
Linear independence
Linear independence
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
Polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
Proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
Pumping lemma
Pumping lemma
In the theory of formal languages in computability theory, a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of...
Simpson's rule
Simpson's rule
In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:...

Articles which mention dependencies of theorems

Cauchy's integral formula
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all...
Cauchy integral theorem
Computational geometry
Computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...
Fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
Lambda calculus
Lambda calculus
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
Invariance of domain
Invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states:The theorem and its proof are due to L.E.J. Brouwer, published in 1912...
Minkowski inequality
Minkowski inequality
In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of Lp...
Nash embedding theorem
Nash embedding theorem
The Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...
Open mapping theorem (functional analysis)
Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map...
Product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
Riemann integral
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The Riemann integral is unsuitable for many theoretical purposes...
Time hierarchy theorem
Time hierarchy theorem
In computational complexity theory, the time hierarchy theorems are important statements about time-bounded computation on Turing machines. Informally, these theorems say that given more time, a Turing machine can solve more problems...
- Deterministic time hierarchy theorem

Theorems of which articles are primarily devoted to proving them

Articles devoted to theorems of which a (sketch of a) proof is given

Articles devoted to algorithms in which their correctness is proven

Articles where example statements are proven

Other articles containing proofs

Articles which mention dependencies of theorems

See also