Number theoretic algorithms | Articles containing proofs
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules,and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers. By or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity. The version of the Euclidean algorithm described above (and by Euclid) can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other. A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two (with this version, the algorithm stops when reaching a zero remainder). With this improvement, the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer. This was proven by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. Additional methods for improving the algorithm's efficiency were developed in the 20th century. The Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. Finally, it can be used as a basic tool for proving theorems in number theory such as Lagrange's four-square theorem and the uniqueness of prime factorizations. The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains. (Wikipedia).
The Euclidean Algorithm: How and Why, Visually
We explain the Euclidean algorithm to compute the gcd, using visual intuition. You'll never forget it once you see the how and why. Then we write it out formally and do an example. This is part of a playlist on GCDs and the Euclidean algorithm: https://www.youtube.com/playlist?list=PLrm
From playlist GCDs and Euclidean algorithm
The extended Euclidean algorithm in one simple idea
An intuitive explanation of the extended Euclidean algorithm as a simple modification of the Euclidean algorithm. This video is part of playlist on GCDs and the Euclidean algorithm: https://www.youtube.com/playlist?list=PLrm9Y---qlNxXccpwYQfllCrHRJWwMky-
From playlist GCDs and Euclidean algorithm
Introduction to Number Theory (Part 4)
The Euclidean algorithm is established and Bezout's theorem is proved.
From playlist Introduction to Number Theory
Euclidean Algorithm - An example ← Number Theory
The Euclidean Algorithm is an efficient method for computing the greatest common divisor of two integers. We demonstrate the algorithm with an example. Teacher: Michael Harrison ♦♦♦♦♦♦♦♦♦♦ Ways to support our channel: ► Join our Patreon : https://www.patreon.com/socratica ► Make a on
From playlist Number Theory
EUCLIDEAN ALGORITHM - DISCRETE MATHEMATICS
Visit our website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW *--Playlists--* Discrete Mathematics 1: https://www.youtube.com/playlist?list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz Discrete Mathematics 2: https://www.youtube.com/playlist?list=PLDDGPdw7e6Aj0amDsYInT_8p6xTS
From playlist Discrete Math 1
[Discrete Mathematics] Euclidean Algorithm and GCDs Examples
We do a question on the Euclidean Algorithm and then tackle a proof about GCDs. LIKE AND SHARE THE VIDEO IF IT HELPED! Visit our website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW *--Playlists--* Discrete Mathematics 1: https://www.youtube.com/playlist?list=PLDDG
From playlist Discrete Math 1
From playlist L. Number Theory
RNT2.3.1. Euclidean Algorithm for Gaussian Integers
Ring Theory: We use the Euclidean algorithm to find the GCD of the Gaussian integers 11+16i and 10+11i. Then we solve for the coefficients in Bezout's identity in this case.
From playlist Abstract Algebra
Introduction to Public-Key Cryptography
From playlist Cryptography Lectures
Could this be the foundation of Number Theory? The Euclidean Algorithm visualized
The Euclidean Algorithm might just be the most fundamental idea in all of Number Theory. In this video I introduce the Euclidean Algorithm, taking inspiration from Martin H. Weissman's An Illustrated Theory of Numbers. You can find Martin's book here: http://illustratedtheoryofnumbers.co
From playlist Summer of Math Exposition Youtube Videos
MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015
Arcane Algorithm Archive: Euclidean Algorithm -- Day 1
Well, we finished an algorithm today. It was quick. -- Watch live at https://www.twitch.tv/simuleios
From playlist Algorithm-archive
Ring Theory: We define Euclidean domains as integral domains with a division algorithm. We show that euclidean domains are PIDs and UFDs, and that Euclidean domains allow for the Euclidean algorithm and Bezout's Identity.
From playlist Abstract Algebra
Boris Springborn: Discrete Uniformization and Ideal Hyperbolic Polyhedra
CATS 2021 Online Seminar Boris Springborn, Technical University of Berlin Abstract: This talk will be about two seemingly unrelated problems: 00:46:00 A discrete version of the uniformization problem for piecewise flat surfaces, and 00:35:48 Constructing ideal hyperbolic polyhedra with p
From playlist Computational & Algorithmic Topology (CATS 2021)
Haotian Jiang: Minimizing Convex Functions with Integral Minimizers
Given a separation oracle SO for a convex function f that has an integral minimizer inside a box with radius R, we show how to find an exact minimizer of f using at most • O(n(n + log(R))) calls to SO and poly(n,log(R)) arithmetic operations, or • O(nlog(nR)) calls to SO and exp(O(n)) · po
From playlist Workshop: Continuous approaches to discrete optimization
BM10.1. The Euclidean Algorithm for the Integers
Basic Methods: Revisiting the proof of Bezout's Identity, we give an algorithm for computing gcd(m, n) without factoring m and n. In turn, the Euclidean algorithm provides a method for finding the coefficients in Bezout's identity.
From playlist Math Major Basics
A Short Course in Algebra and Number Theory - Elementary Number Theory
To supplement a course taught at The University of Queensland's School of Mathematics and Physics I present a very brief summary of algebra and number theory for those students who need to quickly refresh that material or fill in some gaps in their understanding. This is the fourth lectu
From playlist A Short Course in Algebra and Number Theory