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Algorithm for computing greatest common divisors
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers
Euclidean_algorithm
Method for computing the relation of two integers with their greatest common divisor
arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common
Extended_Euclidean_algorithm
Algorithm used for points in euclidean space
in Voronoi diagrams. Although the algorithm may be applied most directly to the Euclidean plane, similar algorithms may also be applied to higher-dimensional
Lloyd's_algorithm
Algorithm for computing the greatest common divisor
The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor
Binary_GCD_algorithm
Greatest common divisor of polynomials
polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Commutative ring with a Euclidean division
generalization of Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the
Euclidean_domain
Largest integer that divides given integers
a) = |a|. This case is important as the terminating step of the Euclidean algorithm. The above definition is unsuitable for defining gcd(0, 0), since
Greatest_common_divisor
Division with remainder of integers
are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental
Euclidean_division
Concept in modular arithmetic
RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm)
Modular multiplicative inverse
Modular_multiplicative_inverse
Theorem about the Euclidean algorithm
is the result of Gabriel Lamé's analysis of the complexity of the Euclidean algorithm. Using Fibonacci numbers, he proved in 1844 that when looking for
Lamé's_theorem
Topics referred to by the same term
numbers Euclidean domain, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and the
Euclidean
Maximally even 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
Euclidean_rhythm
Algorithm for public-key cryptography
λ(n) = lcm(p − 1, q − 1). The lcm may be calculated through the Euclidean algorithm, since lcm(a, b) = |ab|/gcd(a, b). λ(n) is kept secret. Choose
RSA_cryptosystem
Error correction code
popular algorithms for this task are: Peterson–Gorenstein–Zierler algorithm Berlekamp–Massey algorithm Sugiyama Euclidean algorithm Peterson's algorithm is
BCH_code
NP-hard problem in combinatorial optimization
deterministic algorithm and within ( 33 + ε ) / 25 {\displaystyle (33+\varepsilon )/25} by a randomized algorithm. The TSP, in particular the Euclidean variant
Travelling_salesman_problem
Condition under which an odd prime is a sum of two squares
{p}}} . Once x {\displaystyle x} is determined, one can apply the Euclidean algorithm with p {\displaystyle p} and x {\displaystyle x} . Denote the first
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Vector quantization algorithm minimizing the sum of squared deviations
is the minimum Euclidean distance assignment. Hartigan, J. A.; Wong, M. A. (1979). "Algorithm AS 136: A k-Means Clustering Algorithm". Journal of the
K-means_clustering
Ancient Greek mathematician (fl. 300 BC)
numbers and other arithmetic-related concepts. Book 7 includes the Euclidean algorithm, a method for finding the greatest common divisor of two numbers
Euclid
Fast greatest common divisor algorithm
GCD algorithm, named after D. H. Lehmer, is a fast GCD algorithm for multiple-precision arithmetic, which improves on the simpler Euclidean algorithm by
Lehmer's_GCD_algorithm
Algorithm for fast modular multiplication
coprime. It can be constructed using the extended Euclidean algorithm. The extended Euclidean algorithm efficiently determines integers R′ and N′ that satisfy
Montgomery modular multiplication
Montgomery_modular_multiplication
Complex number whose real and imaginary parts are both integers
properties with integers: they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization and
Gaussian_integer
Algorithm for computing Gröbner bases
Gröbner bases. The Euclidean algorithm for computing the polynomial greatest common divisor is a special case of Buchberger's algorithm restricted to polynomials
Buchberger's_algorithm
About simultaneous modular congruences
Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely
Chinese_remainder_theorem
Error-correcting codes
decoding algorithm. In 1975, another improved BCH scheme decoder was developed by Yasuo Sugiyama, based on the extended Euclidean algorithm. In 1977,
Reed–Solomon_error_correction
Overview of and topical guide to algorithms
an algorithm eventually halts Turing machine — mathematical model of computation used in computability theory Euclidean algorithm — ancient algorithm for
Outline_of_algorithms
Mathematical puzzle
problems requiring integer solutions in the 3rd century CE. The Euclidean algorithm for greatest common divisor which underlies the solution of such
The_monkey_and_the_coconuts
Relating two numbers and their greatest common divisor
Bézout coefficients for (a, b); they are not unique. The extended Euclidean algorithm can be used to compute a minimal pair of Bézout coefficients, meaning
Bézout's_identity
Number represented as a0+1/(a1+1/...)
fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number p {\displaystyle
Simple_continued_fraction
Mathematical algorithm
extended Euclidean algorithm. To find the needed a {\displaystyle a} , b {\displaystyle b} , A {\displaystyle A} , and B {\displaystyle B} the algorithm uses
Pollard's rho algorithm for logarithms
Pollard's_rho_algorithm_for_logarithms
Methods of approximating line segments for pixel displays
Euclidean algorithm, as well as Farey sequences and a number of related mathematical constructs. Bresenham's line algorithm Circle drawing algorithm Rasterization
Line_drawing_algorithm
Algebraic structure
computed by using the extended Euclidean algorithm (see Modular multiplicative inverse § Extended Euclidean algorithm). Let F {\displaystyle F} be a finite
Finite_field
Digital verification standard
before the message is known. It may be computed using the extended Euclidean algorithm or using Fermat's little theorem as k q − 2 mod q {\displaystyle
Digital_Signature_Algorithm
Chu–Liu/Edmonds' algorithm): find maximum or minimum branchings Euclidean minimum spanning tree: algorithms for computing the minimum spanning tree of a set of points
List_of_algorithms
Mathematical treatise by Euclid
Euclidean geometry, elementary number theory, and incommensurability. These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm
Euclid's_Elements
Exponentation in modular arithmetic
multiplicative inverse d of b modulo m (for instance by using extended Euclidean algorithm). More precisely: c = be mod m = d−e mod m, where e < 0 and b ⋅ d
Modular_exponentiation
Algorithms which recursively solve subproblems
Babylonia in 200 BC. Another ancient decrease-and-conquer algorithm is the Euclidean algorithm to compute the greatest common divisor of two numbers by
Divide-and-conquer_algorithm
Quantum algorithm for integer factorization
using the Euclidean algorithm. If this produces a nontrivial factor (meaning gcd ( a , N ) ≠ 1 {\displaystyle \gcd(a,N)\neq 1} ), the algorithm is finished
Shor's_algorithm
Mathematical procedure
extension of the Euclidean algorithm can find any integer relation that exists between any two real numbers x1 and x2. The algorithm generates successive
Integer_relation_algorithm
In mathematics, the Jacobi–Perron algorithm is a generalization of the Euclidean algorithm to n-tuples of real numbers, which addresses Hermite's problem
Jacobi–Perron_algorithm
Sequence of operations for a task
described in the Introduction to Arithmetic by Nicomachus, and the Euclidean algorithm, which was first described in Euclid's Elements (c. 300 BC).Examples
Algorithm
Algorithm in computational number theory
Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Algebraic structure
common divisor (although it may not be possible to find it using the Euclidean algorithm). If x and y are elements of a PID without common divisors, then
Principal_ideal_domain
Branch of pure mathematics
number theory, including prime numbers and divisibility. He gave the Euclidean algorithm for computing the greatest common divisor of two numbers and a proof
Number_theory
Method in computational algebra
is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm. With some abstract algebra, the idea behind Berlekamp's algorithm becomes
Berlekamp's_algorithm
Tree representation of the abstract syntactic structure of source code
as concrete syntax tree Semantic resolution tree (SRT) Shunting-yard algorithm Syntax (programming languages) Symbol table TreeDL Abstract Syntax Tree
Abstract_syntax_tree
Indian mathematician and astronomer (598–668)
as Nx2 + 1 = y2 (called Pell's equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the "pulverizer" since it breaks
Brahmagupta
Dijkstra notation with non-deterministic conditionals
b hold the greatest common divisor of A and B. Dijkstra sees in this algorithm a way of synchronizing two infinite cycles a := a - b and b := b - a in
Guarded_Command_Language
Number in {..., –2, –1, 0, 1, 2, ...}
the division of a by b. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. The above says that
Integer
Use of functions that call themselves
the call stack. The iterative algorithm requires a temporary variable, and even given knowledge of the Euclidean algorithm it is more difficult to understand
Recursion_(computer_science)
Algorithm for integer factorization
classes modulo n {\displaystyle n} , performed using the extended Euclidean algorithm. In particular, division by some v mod n {\displaystyle v{\bmod {n}}}
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
Mathematical test in control system theory
Bistritz test. The Routh test can be derived through the use of the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices. Hurwitz derived
Routh–Hurwitz stability criterion
Routh–Hurwitz_stability_criterion
Approximation for the travelling salesman problem
special case of Euclidean space of dimension d {\displaystyle d} , for any c > 0 {\displaystyle c>0} , there is a polynomial-time algorithm that finds a
Christofides_algorithm
Mathematics of Ancient Greece and the Mediterranean, 5th BC to 6th AD
theory, including prime numbers and divisibility. He gave an algorithm, the Euclidean algorithm, for computing the greatest common divisor of two numbers
Ancient_Greek_mathematics
Algorithm for finding shortest paths
path problem. A* search algorithm Bellman–Ford algorithm Euclidean shortest path Floyd–Warshall algorithm Johnson's algorithm Longest path problem Parallel
Dijkstra's_algorithm
Two numbers without shared prime factors
are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm. The number of integers coprime
Coprime_integers
Shortest network connecting points
A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system
Euclidean minimum spanning tree
Euclidean_minimum_spanning_tree
Problem of inverting exponentiation in groups
congruence modulo p {\displaystyle p} in the integers. The extended Euclidean algorithm finds k {\displaystyle k} quickly. With Diffie–Hellman, a cyclic
Discrete_logarithm
Fully simplified fraction
find the greatest common divisor, the Euclidean algorithm or prime factorization can be used. The Euclidean algorithm is commonly preferred because it allows
Irreducible_fraction
Method for division with remainder
result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software. Division algorithms fall into
Division_algorithm
Class of algorithms which use a moving line to solve geometrical problems
various problems in Euclidean space. It is one of the critical techniques in computational geometry. The idea behind algorithms of this type is to imagine
Sweep_line_algorithm
French mathematician (1795–1870)
real number. He is also known for his running time analysis of the Euclidean algorithm, marking the beginning of computational complexity theory. In 1844
Gabriel_Lamé
Algorithm to compute the maximum flow in a network
Ford–Fulkerson algorithm (FFA) is a greedy algorithm that computes the maximum flow in a flow network. It is sometimes called a "method" instead of an "algorithm" as
Ford–Fulkerson_algorithm
Problem of computing shortest paths around geometric obstacles
S2CID 69747. Hershberger, John; Suri, Subhash (1999), "An optimal algorithm for Euclidean shortest paths in the plane", SIAM Journal on Computing, 28 (6):
Euclidean_shortest_path
Amount left over after computation
proof of this result, see Euclidean division. For algorithms describing how to calculate the remainder, see Division algorithm.) The remainder, as defined
Remainder
Public-key cryptosystem
modular multiplicative inverse can be computed using the extended Euclidean algorithm. An alternative is to compute s − 1 {\displaystyle s^{-1}} as c 1
ElGamal_encryption
although the Euclidean algorithm is defined for polynomials with rational coefficients. In fact, in this case, the Euclidean algorithm requires one to
Primitive_part_and_content
'Best' approximation of a function by a rational function of given order
series. One way to compute a Padé approximant is via the extended Euclidean algorithm for the polynomial greatest common divisor. The relation R ( x )
Padé_approximant
topics named after the Greek mathematician Euclid. Euclidean algorithm Extended Euclidean algorithm Euclidean division Euclid–Euler theorem Euclid number Euclid's
List of things named after Euclid
List_of_things_named_after_Euclid
Measure of algorithmic complexity
the operands. Some algorithms run in polynomial time in one model but not in the other one. For example: The Euclidean algorithm runs in polynomial time
Strongly-polynomial_time
Number-theoretic algorithm
replace r0 with m - r0, which will still be a root of -d). Then the Euclidean algorithm can be employed to find r 1 ≡ m ( mod r 0 ) {\displaystyle r_{1}\equiv
Cornacchia's_algorithm
Study of research methods
unambiguous manner for each application. For example, the Euclidean algorithm is an algorithm that solves the problem of finding the greatest common divisor
Methodology
On prime factors of integer products
divisible by n. The following proof is inspired by Euclid's version of Euclidean algorithm, which proceeds by using only subtractions. Suppose that n ∣ a b
Euclid's_lemma
Algorithmic runtime requirements for common math procedures
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Mathematical software
Cantor–Zassenhaus algorithm. Greatest common divisor via e.g. Euclidean algorithm Gaussian elimination Gröbner basis via e.g. Buchberger's algorithm; generalization
Computer_algebra_system
Integer factorization algorithm
{\displaystyle 1649=\gcd(194,1649)\cdot \gcd(34,1649)=97\cdot 17} using the Euclidean algorithm to calculate the greatest common divisor. So the problem has now
Quadratic_sieve
Cryptographic algorithm created by Adi Shamir
B such that A*B % p == 1). This can be computed via the extended Euclidean algorithm http://en.wikipedia.org/wiki/Modular_multiplicative_inverse#Computation
Shamir's_secret_sharing
Algebraic structure
an easy algorithm (such as long division) for computing the Euclidean division. The Euclidean division is the basis of the Euclidean algorithm for polynomials
Polynomial_ring
Computation model defining an abstract machine
the operands. Some algorithms run in polynomial time in one model but not in the other one. For example: The Euclidean algorithm runs in polynomial time
Turing_machine
Numerical eigenvalue calculation
The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the m {\displaystyle m} "most
Lanczos_algorithm
Non-parametric classification method
weighted by the inverse of their distance. This algorithm works as follows: Compute the Euclidean or Mahalanobis distance from the query example to
K-nearest_neighbors_algorithm
Counting polynomial roots in an interval
divisors. This amounts to replacing the remainder sequence of the Euclidean algorithm by a pseudo-remainder sequence, a pseudo remainder sequence being
Sturm's_theorem
Least common multiple Euclidean algorithm Coprime Euclid's lemma Bézout's identity, Bézout's lemma Extended Euclidean algorithm Table of divisors Prime
List_of_number_theory_topics
Algorithm for factoring polynomials over finite fields
field is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm. One important application of the Cantor–Zassenhaus algorithm is in computing
Cantor–Zassenhaus_algorithm
Used to count, measure, and label
primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers. In 240 BC
Number
Number used for counting
by a and b. This Euclidean division is key to the several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number
Natural_number
Inherent difficulty of computational problems
systems. An early example of algorithm complexity analysis is the running time analysis of the Euclidean algorithm done by Gabriel Lamé in 1844. Before
Computational complexity theory
Computational_complexity_theory
Complex number whose mapping on a coordinate plane produces a triangular lattice
} Here, α, β, κ, ρ are all Eisenstein integers. This algorithm implies the Euclidean algorithm, which proves Euclid's lemma and the unique factorization
Eisenstein_integer
Problem of constructing equal-area shapes
difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence
Squaring_the_circle
Computation modulo a fixed integer
Bézout's equation a x + m y = 1 for x, y, by using the Extended Euclidean algorithm. In particular, if p is a prime number, then a is coprime with p
Modular_arithmetic
17th-century conjecture proved by Andrew Wiles in 1994
Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC). Many Diophantine equations have a form similar
Fermat's_Last_Theorem
(Mathematical) decomposition into a product
principal ideal domain, and thus a UFD. In a Euclidean domain, Euclidean division allows defining a Euclidean algorithm for computing greatest common divisors
Factorization
lemma Bellman–Ford algorithm (to do) Euclidean algorithm Kruskal's algorithm Gale–Shapley algorithm Prim's algorithm Shor's algorithm (incomplete) Basis
List_of_mathematical_proofs
Arithmetic operation
integers as result, is sometimes called Euclidean division, because it is the basis of the Euclidean algorithm. Give the integer quotient as the answer
Division_(mathematics)
Rhythmic pattern in Cuban music
and related African bell patterns. Toussaint uses geometry and the Euclidean algorithm as a means of exploring the significance of clave. The most common
Clave_(rhythm)
Minimum spanning forest algorithm that greedily adds edges
Kruskal's algorithm finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree
Kruskal's_algorithm
Representation of modular integers by "small" fractions
extended Euclidean algorithm, allows us to provide a proof that leads to an efficient algorithm that has the same computational complexity of the Euclidean algorithm
Thue's_lemma
Family of RISC-based computer architectures
skipped instruction. An algorithm that provides a good example of conditional execution is the subtraction-based Euclidean algorithm for computing the greatest
Arm_architecture_family
Class of algorithms that find approximate solutions to optimization problems
improved understanding, the algorithms may be refined to become more practical. One such example is the initial PTAS for Euclidean TSP by Sanjeev Arora (and
Approximation_algorithm
French mathematician (1795–1838)
of them seems to be the first to recognize the worst case in the euclidean algorithm: when the inputs are proportional to consecutive Fibonacci numbers
Émile_Léger
Mathematical algorithm
Kuṭṭaka algorithm has much similarity with and can be considered as a precursor of the modern day extended Euclidean algorithm. The latter algorithm is a
Kuṭṭaka
EUCLIDEAN ALGORITHM
EUCLIDEAN ALGORITHM
EUCLIDEAN ALGORITHM
EUCLIDEAN ALGORITHM
Girl/Female
Polish
noble.
Girl/Female
Tamil
Universe
Surname or Lastname
English (East Anglia)
English (East Anglia) : unexplained.
Boy/Male
Gujarati, Hindu, Indian
Announce
Girl/Female
American, Australian, Danish, French, Latin
Ready for Battle; Armoured; Warrior Woman
Boy/Male
Latin
Admonishes.
Girl/Female
Celtic
Renowned friend. Feminine of Marvin: Lives by the Sea.
Boy/Male
American, British, English, German
Mighty Spearman; Strong
Girl/Female
Australian, Danish, Latin
Dark; The Adriatic Sea Region; From Adria
Female
Scandinavian
Scandinavian form of Old Norse SigrÃðr, SIGRID means "beautiful victory."
EUCLIDEAN ALGORITHM
EUCLIDEAN ALGORITHM
EUCLIDEAN ALGORITHM
EUCLIDEAN ALGORITHM
EUCLIDEAN ALGORITHM
n.
Alt. of Algorithm
n.
The surface of constant negative curvature generated by the revolution of a tractrix. This surface corresponds in non-Euclidian space to the sphere in ordinary space. An important property of the surface is that any figure drawn upon it can be displaced in any way without tearing it or altering in size any of its elements.
n.
Related to Euclid, or to the geometry of Euclid.
n.
The art of calculating by nine figures and zero.
n.
The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc.