"an efficient method for computing the greatest common divisor (GCD) of two integers"
An algorithm for finding the GCD of two numbers.
Greatest Common Divisor (GCD): The GCD of two integers is the largest positive integer that divides both of them without leaving a remainder. The Euclidean algorithm is used to find the GCD of two integers.
Division algorithm: The Division algorithm states that given two non-negative integers "a" and "b", there exist unique integers "q" and "r" such that "a=bq+r" and 0<=r
Extended Euclidean algorithm: The Extended Euclidean algorithm is used to solve the Bezout's identity, which states that for any two integers "a" and "b", there exist integers "x" and "y" such that ax+by=GCD(a,b).
Modular arithmetic: Modular arithmetic is a special type of arithmetic, where numbers are considered to be congruent modulo some integer "m". It is used in the Euclidean algorithm to simplify computations.
Prime factorization: A prime factorization of an integer is a way of expressing it as a product of prime numbers. The Euclidean algorithm is used to find the GCD of two integers, which is useful for finding the prime factorization.
Relatively prime numbers: Two numbers are relatively prime if they have no common factors other than 1. This concept is important in understanding the Euclidean algorithm and its application.
Diophantine equations: A Diophantine equation is an equation in which only integer solutions are allowed. The Extended Euclidean algorithm is often used to find solutions to Diophantine equations.
Chinese Remainder Theorem: The Chinese Remainder Theorem states that if we have a system of congruences modulo pairwise coprime integers, then there exists a unique solution modulo their product.
Euclidean domain: An integral domain in which the Euclidean algorithm can be used to compute the GCD of any two elements. This concept is important in understanding the Euclidean algorithm and its use in modern number theory.
Bezout's theorem: Bezout's theorem is a more general version of Bezout's identity, which applies to any two elements of a Euclidean domain.
Basic Euclidean algorithm: The basic form of the algorithm involves finding the greatest common divisor (GCD) of two integers using repeated division and recursion.
Extended Euclidean algorithm: The extended version not only calculates the GCD of two integers but also returns two integers x and y that satisfy the equation ax + by = gcd(a, b).
Binary Euclidean algorithm: Similar to the basic algorithm, the binary version involves repeated division and recursion, but it uses binary representation of the integers to speed up the process.
Euclidean subspace algorithm: This algorithm is used in linear algebra to compute the GCD of polynomials and matrices.
Euclidean vector algorithm: This algorithm is used in linear algebra to compute the GCD of vectors.
Diophantine equation algorithm: It is a generalization of the extended Euclidean algorithm used to solve linear Diophantine equations.
Modular inverse algorithm: The algorithm finds the multiplicative inverse of an integer modulo m using the extended Euclidean algorithm.
Chinese remainder theorem algorithm: This algorithm uses the extended Euclidean algorithm to solve systems of linear congruences.
Continued fraction algorithm: This algorithm uses continued fractions to approximate the GCD of two integers.
Quadratic sieve algorithm: This is an advanced factorization algorithm that utilizes Euclid's algorithm to find the GCD of two polynomials.
"named after the ancient Greek mathematician Euclid"
"step-by-step procedure for performing a calculation according to well-defined rules"
"one of the oldest algorithms in common use"
"reduce fractions to their simplest form", "part of many other number-theoretic and cryptographic calculations"
"the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number"
"Yes, the GCD can be expressed as a linear combination of the two original numbers"
"the GCD can always be expressed in this way"
"replacing the larger of the two numbers by its remainder when divided by the smaller of the two"
"the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer"
"proven by Gabriel Lamé in 1844 (Lamé's Theorem)"
"used to secure internet communications", "breaking these cryptosystems by factoring large composite numbers"
"Yes, it may be used to solve Diophantine equations"
"other types of numbers, such as Gaussian integers and polynomials of one variable"
"led to modern abstract algebraic notions such as Euclidean domains"
"Computations using this algorithm form part of the cryptographic protocols"
"Lagrange's four-square theorem and the uniqueness of prime factorizations" Apologies, but I am not able to continue this response as the available content is limited.