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243 relations: Absorbing element, AF+BG theorem, AKS primality test, Algebraic integer, Algebraic-group factorisation algorithm, Algebraically closed field, Algorithm, Algorithm characterizations, Amicable numbers, Anomalous cancellation, Aperiodic graph, Arbitrary-precision arithmetic, Arithmetic billiards, Arithmetic function, Associative property, Bézout domain, Bézout's identity, Beal conjecture, Berlekamp's algorithm, Bhargava factorial, Binary GCD algorithm, Binomial coefficient, Birkhoff's representation theorem, Blum Blum Shub, Boolean algebra, Boolean algebra (structure), Brigitte Vallée, Buchberger's algorithm, Calkin–Wilf tree, Cancelling out, Cantor–Zassenhaus algorithm, Certifying algorithm, CoffeeScript, Coin problem, Complete lattice, Computable function, Computational complexity of mathematical operations, Computer algebra system, Congruence of squares, Continued fraction, Coppersmith's attack, Coprime integers, Coxeter notation, Coxeter–Dynkin diagram, Crisscross method, Cubic reciprocity, Cycle detection, Cyclic group, Cyclotomic polynomial, Dc (computer program), ..., Diophantine equation, Dirichlet character, Distributive lattice, Distributive property, Divide and conquer algorithm, Division (mathematics), Division algorithm, Divisor, Divisor sum identities, Dixon's factorization method, ECL programming language, Eisenstein's criterion, ElGamal signature scheme, Equivalence relation, Erdős–Woods number, Euclid, Euclid's Elements, Euclidean algorithm, Euclidean division, Euclidean domain, Euclidean rhythm, Eugene McDonnell, Euler pseudoprime, Euler's totient function, Euler–Fokker genus, Euler–Mascheroni constant, Extended Euclidean algorithm, Factorization, Factorization of polynomials, Feit–Thompson conjecture, Fibonacci number, Formula for primes, Fraction (mathematics), Free abelian group, Friendly number, Fundamental discriminant, Fundamental frequency, Fundamental theorem of arithmetic, Gabriel Lamé, Gauss's lemma (polynomial), Gaussian integer, GCD, GCD domain, GCD test, GCF, GCM, Gear, General number field sieve, Generalized Clifford algebra, Generalized continued fraction, Generating set of a group, GiNaC, Glossary of commutative algebra, Godfried Toussaint, Group (mathematics), Guarded Command Language, HCF, Heath-Brown–Moroz constant, Heronian triangle, Hilbert's tenth problem, Homotopy groups of spheres, HP-32S, Identity element, Idoneal number, In-place matrix transposition, Integer, Integer factorization, Integer-valued polynomial, Irrationality sequence, Irreducible fraction, Irreducible polynomial, Jacobi symbol, Java Platform, Standard Edition, Kasiski examination, Kempner function, Knapsack problem, Kuṭṭaka, Large numbers, Lattice (order), Lattice reduction, Least common divisor, Least common multiple, Lehmer's GCD algorithm, Lenstra elliptic-curve factorization, List of acronyms: G, List of acronyms: H, List of algorithms, List of mathematical abbreviations, List of mathematical symbols, List of mathematical symbols by subject, List of number theory topics, List of numbers, List of terms relating to algorithms and data structures, Lonely runner conjecture, Lowest common denominator, Lowest common divisor, Lowest common factor, Lucas pseudoprime, Macsyma, Maple (software), Markov chain, Math symbol parentheses, Maximal and minimal elements, Maximal common divisor, Method of successive substitution, Metric modulation, Miller index, Miller–Rabin primality test, Missing fundamental, Modular multiplicative inverse, Modular representation theory, Multiplicative function, Multiplicative order, Murderous Maths, National Academic League, Necklace polynomial, Necklace ring, Novo Aeon, Number, Number theory, Numerical semigroup, Numerically controlled oscillator, Opal (programming language), Order (group theory), Order theory, Outline of arithmetic, P (complexity), P-complete, Perfect power, Perron–Frobenius theorem, Pitch (music), Pollard's p − 1 algorithm, Polygram (geometry), Polynomial Diophantine equation, Polynomial greatest common divisor, Polynomial ring, Porter's constant, Positional notation, Primality test, Primefree sequence, Primitive part and content, Principal ideal, Principal ideal domain, Proof of Fermat's Last Theorem for specific exponents, Pythagorean quadruple, Pythagorean theorem, Quadratic Gauss sum, Quantum revival, Quartic reciprocity, Rabin cryptosystem, Ramanujan's sum, Rational number, Rational root theorem, Rational sieve, Recursion (computer science), Red auxiliary number, Reduced residue system, Refactorable number, Regular polygon, Representation theory, Resultant, Riesel number, Ring theory, Root of unity, Rotational symmetry, RSA (cryptosystem), Secret sharing using the Chinese remainder theorem, Shanks's square forms factorization, Shellsort, Shor's algorithm, Sierpinski number, Smith normal form, Special number field sieve, Squaring the square, Star of David theorem, Supernatural number, Sylver coinage, SymPy, Systolic array, Table of prime factors, The Perennial Philosophy, Time complexity, Torus knot, Transposable integer, Turk's head knot, Uniform polyhedron compound, Unique factorization domain, Unit fraction, Use-define chain, Wieferich prime, Z-group, Zolotarev's lemma, 0.999.... 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Absorbing element
In mathematics, an absorbing element is a special type of element of a set with respect to a binary operation on that set.
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AF+BG theorem
In algebraic geometry, a field of mathematics, the AF+BG theorem (also known as Max Noether's fundamental theorem) is a result of Max Noether which describes when the equation of an algebraic curve in the complex projective plane can be written in terms of the equations of two other algebraic curves.
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AKS primality test
The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled "PRIMES is in P".
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Algebraic integer
In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).
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Algebraic-group factorisation algorithm
Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors p1, p2,...
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Algebraically closed field
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.
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Algorithm
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.
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Algorithm characterizations
Algorithm characterizations are attempts to formalize the word algorithm.
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Amicable numbers
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number.
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Anomalous cancellation
An anomalous cancellation or accidental cancellation is a particular kind of arithmetic procedural error that gives a numerically correct answer.
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Aperiodic graph
In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph.
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Arbitrary-precision arithmetic
In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system.
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Arithmetic billiards
In recreational mathematics, arithmetic billiards provide a geometrical method to determine the least common multiple and the greatest common divisor of two natural numbers by making use of reflections inside a rectangle whose sides are the two given numbers.
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Arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers.
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Associative property
In mathematics, the associative property is a property of some binary operations.
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Bézout domain
In mathematics, a Bézout domain is a form of a Prüfer domain.
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Bézout's identity
In elementary number theory, Bézout's identity (also called Bézout's lemma) is the following theorem: The integers x and y are called Bézout coefficients for (a, b); they are not unique.
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Beal conjecture
The Beal conjecture is the following conjecture in number theory: Equivalently, The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's last theorem.
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Berlekamp's algorithm
In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields).
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Bhargava factorial
In mathematics, Bhargava's factorial function, or simply Bhargava factorial, is a certain generalization of the factorial function developed by the Fields Medal winning mathematician Manjul Bhargava as part of his thesis in Harvard University in 1996.
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Binary GCD algorithm
The binary GCD algorithm, also known as Stein's algorithm, is an algorithm that computes the greatest common divisor of two nonnegative integers.
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Binomial coefficient
In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient.
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Birkhoff's representation theorem
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets.
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Blum Blum Shub
Blum Blum Shub (B.B.S.) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub that is derived from Michael O. Rabin's oblivious transfer mapping.
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Boolean algebra
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.
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Boolean algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.
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Brigitte Vallée
Brigitte Vallée (born June 6, 1950, in Courbevoie, Hauts-de-Seine, France) is a French mathematician and computer scientist.
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Buchberger's algorithm
In computational algebraic geometry and computational commutative algebra, Buchberger's algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order.
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Calkin–Wilf tree
In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond 1-for-1 to the positive rational numbers.
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Cancelling out
Cancelling out is a mathematical process used for removing subexpressions from a mathematical expression, when this removal does not change the meaning or the value of the expression because the subexpressions have equal and opposing effects.
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Cantor–Zassenhaus algorithm
In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields).
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Certifying algorithm
In theoretical computer science, a certifying algorithm is an algorithm that outputs, together with a solution to the problem it solves, a proof that the solution is correct.
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CoffeeScript
CoffeeScript is a programming language that transcompiles to JavaScript.
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Coin problem
With only 2 pence and 5 pence coins, one cannot make 3 pence, but one can make any higher integral amount.
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Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).
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Computable function
Computable functions are the basic objects of study in computability theory.
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Computational complexity of mathematical operations
The following tables list the computational complexity of various algorithms for common mathematical operations.
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Computer algebra system
A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.
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Congruence of squares
In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms.
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Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.
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Coppersmith's attack
Coppersmith's attack describes a class of cryptographic attacks on the public-key cryptosystem RSA based on the Coppersmith method.
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Coprime integers
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
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Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups.
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Coxeter–Dynkin diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).
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Crisscross method
The Thorpe's Crisscross method is a method of finding out the chemical formula of a metal and non-metal that combine to form an ionic bond created by high school chemistry teacher Kim Thorpe.
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Cubic reciprocity
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is solvable.
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Cycle detection
In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Cyclotomic polynomial
In mathematics, more specifically in algebra, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any.
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Dc (computer program)
dc (desk calculator) is a cross-platform reverse-polish calculator which supports arbitrary-precision arithmetic.
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Diophantine equation
In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values).
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Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.
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Distributive lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other.
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Distributive property
In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.
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Divide and conquer algorithm
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion.
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Division (mathematics)
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.
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Division algorithm
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of division.
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Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.
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Divisor sum identities
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number n, or equivalently the Dirichlet convolution of an arithmetic function f(n) with one: These identities include applications to sums of an arithmetic function over just the proper prime divisors of n. We also define periodic variants of these divisor sums with respect to the greatest common divisor function in the form of Well-known inversion relations that allow the function f(n) to be expressed in terms of g(n) are provided by the Mobius inversion formula.
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Dixon's factorization method
In number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method.
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ECL programming language
The ECL programming language and system were an extensible high-level programming language and development environment developed at Harvard University in the 1970s.
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Eisenstein's criterion
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorable into the product of non-constant polynomials with rational coefficients.
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ElGamal signature scheme
The ElGamal signature scheme is a digital signature scheme which is based on the difficulty of computing discrete logarithms.
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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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Erdős–Woods number
In number theory, a positive integer is said to be an Erdős–Woods number if it has the following property: there exists a positive integer such that in the sequence of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints.
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Euclid
Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".
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Euclid's Elements
The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.
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Euclidean algorithm
. EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION.
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Euclidean division
In arithmetic, Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor.
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Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers.
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Euclidean rhythm
The Euclidean rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The Euclidean Algorithm Generates Traditional Musical Rhythms".
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Eugene McDonnell
Eugene Edward McDonnell (October 18, 1926 – August 17, 2010) was a computer science pioneer and long-time contributor to the programming language siblings APL and J. He was a graduate of Brooklyn Technical High School.
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Euler pseudoprime
In arithmetic, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and (where mod refers to the modulo operation).
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Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.
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Euler–Fokker genus
In music theory and tuning, an Euler–Fokker genus (plural: genera), named after Leonhard Euler and Adriaan Fokker,Rasch, Rudolph (2000).
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Euler–Mascheroni constant
The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.
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Extended Euclidean algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs.
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Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
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Factorization of polynomials
In mathematics and computer algebra, factorization of polynomials or polynomial factorization is the process of expressing a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain.
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Feit–Thompson conjecture
In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by.
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Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones: Often, especially in modern usage, the sequence is extended by one more initial term: By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.
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Formula for primes
In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception.
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Fraction (mathematics)
A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.
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Free abelian group
In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis.
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Friendly number
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself.
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Fundamental discriminant
In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms.
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Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform.
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Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
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Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity.
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Gauss's lemma (polynomial)
In algebra, in the theory of polynomials (a subfield of ring theory), Gauss's lemma is either of two related statements about polynomials with integer coefficients.
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Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.
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GCD
GCD may refer to.
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GCD domain
In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD).
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GCD test
In compiler theory of computer science, A Greatest common divisor Test is the test used in study of loop optimization and loop dependence analysis to test the dependency between loop statements.
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GCF
GCF may refer to.
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GCM
GCM may refer to.
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Gear
A gear or cogwheel is a rotating machine part having cut like teeth, or cogs, which mesh with another toothed part to transmit torque.
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General number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than.
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Generalized Clifford algebra
In mathematics, a Generalized Clifford algebra (GCA) is an associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these '''clock-and-shift''' operators introduced by J. J. Sylvester (1882), and organized by Cartan (1898) and Schwinger.
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Generalized continued fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values.
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Generating set of a group
In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.
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GiNaC
GiNaC is a free computer algebra system released under the GNU General Public License.
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Glossary of commutative algebra
This is a glossary of commutative algebra.
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Godfried Toussaint
Godfried T. Toussaint is a Canadian Computer Scientist, a Professor of Computer Science, and the Head of the Computer Science Program at New York University Abu Dhabi (NYUAD) in Abu Dhabi, United Arab Emirates.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Guarded Command Language
The Guarded Command Language (GCL) is a language defined by Edsger Dijkstra for predicate transformer semantics.
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HCF
HCF may refer to.
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Heath-Brown–Moroz constant
The Heath-Brown–Moroz constant C, named for Roger Heath-Brown and Boris Moroz, is defined as where p runs over the primes.
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Heronian triangle
In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers.
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Hilbert's tenth problem
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900.
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Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other.
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HP-32S
The HP-32S was a programmable RPN Scientific Calculator introduced by Hewlett-Packard in 1988.
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Identity element
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.
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Idoneal number
In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime, prime power, twice one of these, or a power of 2.
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In-place matrix transposition
In-place matrix transposition, also called in-situ matrix transposition, is the problem of transposing an N×M matrix in-place in computer memory, ideally with ''O''(1) (bounded) additional storage, or at most with additional storage much less than NM.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.
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Integer-valued polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true.
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Irrationality sequence
In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers, the sum of the series exists (that is, it converges) and is an irrational number.
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Irreducible fraction
An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered).
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Irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.
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Jacobi symbol
Jacobi symbol for various k (along top) and n (along left side).
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Java Platform, Standard Edition
Java Platform, Standard Edition (Java SE) is a computing platform for development and deployment of portable code for desktop and server environments.
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Kasiski examination
In cryptanalysis, Kasiski examination (also referred to as Kasiski's test or Kasiski's method) is a method of attacking polyalphabetic substitution ciphers, such as the Vigenère cipher.
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Kempner function
In number theory, the Kempner function S(n)Called the Kempner numbers in the Online Encyclopedia of Integer Sequences: see is defined for a given positive integer n to be the smallest number s such that n divides the factorial s!.
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Knapsack problem
The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.
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Kuṭṭaka
Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations.
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Large numbers
Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions.
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Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.
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Lattice reduction
In mathematics, the goal of lattice basis reduction is given an integer lattice basis as input, to find a basis with short, nearly orthogonal vectors.
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Least common divisor
The phrase least common divisor is a confusion of the following two distinct concepts in arithmetic.
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Least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.
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Lehmer's GCD algorithm
Lehmer's GCD algorithm, named after Derrick Henry Lehmer, is a fast GCD algorithm, an improvement on the simpler but slower Euclidean algorithm.
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Lenstra elliptic-curve factorization
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves.
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List of acronyms: G
(Main list of acronyms).
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List of acronyms: H
(Main list of acronyms).
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List of algorithms
The following is a list of algorithms along with one-line descriptions for each.
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List of mathematical abbreviations
This article is a listing of abbreviated names of mathematical functions, function-like operators and other mathematical terminology.
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List of mathematical symbols
This is a list of symbols used in all branches of mathematics to express a formula or to represent a constant.
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List of mathematical symbols by subject
This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic.
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List of number theory topics
This is a list of number theory topics, by Wikipedia page.
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List of numbers
This is a list of articles about numbers (not about numerals).
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List of terms relating to algorithms and data structures
The NIST Dictionary of Algorithms and Data Structures is a reference work maintained by the U.S. National Institute of Standards and Technology.
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Lonely runner conjecture
In number theory, and especially the study of diophantine approximation, the lonely runner conjecture is a conjecture originally due to J. M. Wills in 1967.
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Lowest common denominator
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions.
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Lowest common divisor
The lowest common divisor of any numbers is always 1, making it a useless term, often mistakenly used when one actually wishes to refer to either.
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Lowest common factor
Lowest common factor may refer to the following mathematical terms.
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Lucas pseudoprime
Lucas pseudoprimes and Fibonacci pseudoprimes are composite integers that pass certain tests which all primes and very few composite numbers pass: in this case, criteria relative to some Lucas sequence.
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Macsyma
Macsyma (Project MAC’s SYmbolic MAnipulator) is one of the oldest general purpose computer algebra systems which is still widely used.
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Maple (software)
Maple is a symbolic and numeric computing environment, and is also a multi-paradigm programming language.
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Markov chain
A Markov chain is "a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event".
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Math symbol parentheses
Parentheses in mathematical equations may refer to.
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Maximal and minimal elements
In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.
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Maximal common divisor
In abstract algebra, particularly ring theory, maximal common divisors are an abstraction of the number theory concept of greatest common divisor (GCD).
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Method of successive substitution
In modular arithmetic, the method of successive substitution is a method of solving problems of simultaneous congruences by using the definition of the congruence equation.
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Metric modulation
In music, metric modulation is a change in pulse rate (tempo) and/or pulse grouping (subdivision) which is derived from a note value or grouping heard before the change.
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Miller index
Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices.
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Miller–Rabin primality test
The Miller–Rabin primality test or Rabin–Miller primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay–Strassen primality test.
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Missing fundamental
A harmonic sound is said to have a missing fundamental, suppressed fundamental, or phantom fundamental when its overtones suggest a fundamental frequency but the sound lacks a component at the fundamental frequency itself.
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Modular multiplicative inverse
In mathematics, in particular the area of number theory, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus.
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Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic.
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Multiplicative function
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1).
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Multiplicative order
In number theory, given an integer a and a positive integer n with gcd(a,n).
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Murderous Maths
Murderous Maths is a series of British educational books by author Kjartan Poskitt.
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National Academic League
The National Academic League (NAL) is a popular in junior high schools (middle schools) around the United States.
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Necklace polynomial
In combinatorial mathematics, the necklace polynomials, or (Moreau's) necklace-counting function are the polynomials in α such that By Möbius inversion they are given by where is the classic Möbius function.
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Necklace ring
In mathematics, the necklace ring is a ring introduced by.
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Novo Aeon
Novo Aeon (Portuguese for New Aeon) is the third studio solo album by Brazilian musician Raul Seixas.
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Number
A number is a mathematical object used to count, measure and also label.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Numerical semigroup
In mathematics, a numerical semigroup is a special kind of a semigroup.
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Numerically controlled oscillator
A numerically controlled oscillator (NCO) is a digital signal generator which creates a synchronous (i.e. clocked), discrete-time, discrete-valued representation of a waveform, usually sinusoidal.
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Opal (programming language)
OPAL (OPtimized Applicative Language) is a functional programming language first developed at the Technical University of Berlin.
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Order (group theory)
In group theory, a branch of mathematics, the term order is used in two unrelated senses.
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Order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.
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Outline of arithmetic
Arithmetic is an elementary branch of mathematics that is used by almost everyone for tasks ranging from simple day-to-day counting to advanced science and business calculations.
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P (complexity)
In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class.
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P-complete
In complexity theory, a decision problem is P-complete (complete for the complexity class '''P''') if it is in P and every problem in P can be reduced to it by an appropriate reduction.
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Perfect power
In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer.
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Perron–Frobenius theorem
In linear algebra, the Perron–Frobenius theorem, proved by and, asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices.
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Pitch (music)
Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, or more commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies.
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Pollard's p − 1 algorithm
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974.
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Polygram (geometry)
In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides, so a regular pentagram,, has 5 sides, and the regular hexagram, or 2 has 6 sides divided into two triangles.
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Polynomial Diophantine equation
In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation for which one seeks solutions restricted to be polynomials in the indeterminate.
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Polynomial greatest common divisor
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials.
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Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
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Porter's constant
In mathematics, Porter's constant C arises in the study of the efficiency of the Euclidean algorithm.
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Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers.
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Primality test
A primality test is an algorithm for determining whether an input number is prime.
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Primefree sequence
In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers.
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Primitive part and content
In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients.
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Principal ideal
In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept.
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Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.
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Proof of Fermat's Last Theorem for specific exponents
Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proved by Andrew Wiles in 1995.
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Pythagorean quadruple
A Pythagorean quadruple is a tuple of integers,, and, such that.
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Pythagorean theorem
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.
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Quadratic Gauss sum
In number theory, quadratic Gauss sums are certain finite sums of roots of unity.
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Quantum revival
In quantum mechanics, the quantum revival is a periodic recurrence of the quantum wave function from its original form during the time evolution either many times in space as the multiple scaled fractions in the form of the initial wave function (fractional revival) or approximately or exactly to its original form from the beginning (full revival).
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Quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x4 ≡ p (mod q) to that of x4 ≡ q (mod p).
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Rabin cryptosystem
The Rabin cryptosystem is an asymmetric cryptographic technique, whose security, like that of RSA, is related to the difficulty of factorization.
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Ramanujan's sum
In number theory, a branch of mathematics, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula: where (a, q).
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Rational root theorem
In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation with integer coefficients.
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Rational sieve
In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors.
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Recursion (computer science)
Recursion in computer science is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem (as opposed to iteration).
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Red auxiliary number
In the study of ancient Egyptian mathematics, red auxiliary numbers were additive numbers that summed to a numerator used in Middle Kingdom arithmetic problems.
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Reduced residue system
Any subset R of the integers is called a reduced residue system modulo n if.
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Refactorable number
A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that \tau(n)|n.
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Regular polygon
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
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Resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients).
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Riesel number
In mathematics, a Riesel number is an odd natural number k for which the integers of the form k·2n − 1 are composite for all natural numbers n. In other words, when k is a Riesel number, all members of the following set are composite: In 1956, Hans Riesel showed that there are an infinite number of integers k such that k·2n − 1 is not prime for any integer n.
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Ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.
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Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
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Rotational symmetry
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.
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RSA (cryptosystem)
RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission.
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Secret sharing using the Chinese remainder theorem
Secret sharing consists of recovering a secret S from a set of shares, each containing partial information about the secret.
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Shanks's square forms factorization
Shanks's square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method.
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Shellsort
Shellsort, also known as Shell sort or Shell's method, is an in-place comparison sort.
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Shor's algorithm
Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm that runs on a quantum computer) for integer factorization formulated in 1994.
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Sierpinski number
In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that k \times 2^n + 1 is composite, for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.
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Smith normal form
In mathematics, the Smith normal form is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID).
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Special number field sieve
In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm.
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Squaring the square
Squaring the square is the problem of tiling an integral square using only other integral squares.
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Star of David theorem
The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients.
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Supernatural number
In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers.
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Sylver coinage
Sylver coinage is a mathematical game for two players, invented by John H. Conway.
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SymPy
SymPy is a Python library for symbolic computation.
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Systolic array
In parallel computer architectures, a systolic array is a homogeneous network of tightly coupled data processing units (DPUs) called cells or nodes.
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Table of prime factors
The tables contain the prime factorization of the natural numbers from 1 to 1000.
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The Perennial Philosophy
The Perennial Philosophy is a comparative study of mysticism by the British writer and novelist Aldous Huxley.
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Time complexity
In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.
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Torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3.
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Transposable integer
The digits of some specific integers permute or shift cyclically when they are multiplied by a number n. Examples are.
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Turk's head knot
A Turk's head knot is a decorative knot with a variable number of interwoven strands, forming a closed loop.
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Uniform polyhedron compound
A uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.
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Unique factorization domain
In mathematics, a unique factorization domain (UFD) is an integral domain (a non-zero commutative ring in which the product of non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers.
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Unit fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer.
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Use-define chain
A Use-Definition Chain (UD Chain) is a data structure that consists of a use, U, of a variable, and all the definitions, D, of that variable that can reach that use without any other intervening definitions.
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Wieferich prime
In number theory, a Wieferich prime is a prime number p such that p2 divides, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides.
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Z-group
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups.
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Zolotarev's lemma
In number theory, Zolotarev's lemma states that the Legendre symbol for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation: where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplication by a. For example, take a.
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0.999...
In mathematics, 0.999... (also written 0., among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it).
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References
[1] https://en.wikipedia.org/wiki/Greatest_common_divisor
