# Generalizing Euclid"s algorithm, via the regular and Moebius knot trees, order-n arithmetics

by A. G. Schaake

Publisher: Waikato Polytechnic in [Waikato, N.Z

Written in English

## Subjects:

• Euclidean algorithm,
• Knot theory,
• Braid theory

## Edition Notes

Includes bibliographical references (p. 61).

Classifications The Physical Object Statement A.G. Schaake, J.C. Turner. Series Research report ;, no. 196, Research report (University of Waikato. Dept. of Mathematics and Statistics) ;, no. 196. Contributions Turner, J. C. 1928- LC Classifications QA242 .S33 1990 Pagination 61 p. : Number of Pages 61 Open Library OL830651M LC Control Number 95104591

The greatest common divisor (gcd) of two positive integers is the largest integer that divides both without remainder. Euclid’s algorithm is based on the following property: if p>q then the gcd of p and q is the same as the gcd of p%q and q. p%q is the remainder of . Euclids Algorithm Calculator,Euclids Extended Algorithm Calculator. Euclids Algorithm and Euclids Extended Algorithm Video. Looking at Euclid's algorithm for the "Greatest common divisor of two numbers", I'm trying to divine the big-O cpu time for numbers K, and N. Can anybody help? This is the algorithm as I understand it.. Where: max(A,B) = the greater of A or B such that: min(10,3) = 10; min(A,B) = the smaller of A or B such that: min(10,3) = 3.   The Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor of two integers.

Change Euclid to gcd: Lower case initial letter for function names. gcd is the common term for greatest common divisor, the Euclidean algorithm is just one possible implementation. And you are not implementing the Euclidean algorithm. Rename the second loop iterator j to i. I have a question about the Euclid's Algorithm for finding greatest common divisors. gcd(p,q) where p > q and q is a n-bit integer. I'm trying to follow a time complexity analysis on the algorithm . The actual start of the algorithm is not determined, because a and b could be the result of some previous steps in the algorithm. We can write them as r 1 and r 2, with r 1 >=r 2. If r 1 =r 2, then the gcd is r 1 (or r 2). For instance, the gcd(r,r)=r, because clearly r divides itself, . The gcd of two integers can be found by repeated application of the division algorithm, this is known as the Euclidean Algorithm. I.e.: To find the gcd of 81 and 57 by the Euclidean Algorithm, we proceed as follows: 81 = 1 * 57 + 24 57 = 2 * 24 + 9 24 = 2 * 9 + 6 9 = 1 * 6 + 3 6 = 2 * 3 + 0. We repeatedly divide the divisor by the remainder until the remainder is 0.

In this video, we're going to discover together the famous Euclid's Algorithm. Which is, once again, let me remind you, an efficient algorithm for computing the greatest common divisor of two integers. It is based on the following, simple but crucial Euclid's Lemma. An integer d divides both integers a and b, if and only if, d divides a minus b. Definition: An alternate approach to speeding up euclids algorithm is due to lehmer. One notices that when a and b have the same size, the integer part w of the a/b is often single digit. Assume that a,b are very big numbers, ^a,^b are small numbers such that: a/b = ^a/^b then the sequence of quotients produced by EA(a,b) and by. Here's intuitive understanding of runtime complexity of Euclid's algorithm. The formal proofs are covered in various texts such as Introduction to Algorithms and TAOCP Vol 2. First think about what if we tried to take gcd of two Fibonacci numbers F(k+1) and F(k). You might quickly observe that Euclid's algorithm iterates on to F(k) and F(k-1). See also binary GCD, extended Euclid's algorithm, Ferguson-Forcade algorithm. Note: After [CLR90, page ]. Author: PEB. Implementation Worst-case behavior annotated for real time (WOOP/ADA). Go to the Dictionary of Algorithms and Data Structures home page.

## Generalizing Euclid"s algorithm, via the regular and Moebius knot trees, order-n arithmetics by A. G. Schaake Download PDF EPUB FB2

In Eudlid’s Elements, Book VII, proposition 2, (circa B.C.) there appears a method for determining the greatest common divisor of two integers, which is now known simply as Euclid’s algorithm. Schaake, A. and Turner, J. Generalizing Euclid’s Algorithm, via the Regular and Moebius Knot Trees.

Order n-Arithmetics Research Cited by: 1. Euclid's Algorithm. Euclid's algorithm is a famous procedure for finding the gcd, i.e., greatest common divisor (factor) of two integers.

The idea is pretty simple. If N = M×s, with N, M, s, positive integers, then any divisor of M is also a divisor of N, making M their greatest common divisor. If N = M×s then M = gcd(N, M). when N = M×s + R (where all four are positive integers), then. The total number of steps in Euclid’s algorithm cannot exceed five times the number of digits in the smallest of the two numbers.

Two thousands years passed between Euclid’s formulation of his algorithm in around BC, and this proof was given by Gabriel Lamé in The above result is the basis of an eﬃcient algorithm for computing greatest common divisors.

It was described by Euclid around BC in his book the Elements in Propositions 1 and 2 of Book VII. Algorithm (Euclid’s algorithm). Input: a,b ∈ N such that a ≥ b and b 6= 0. Output: gcd(a,b). Procedure: write a = bq+r where 0 ≤ r. Euclidean algorithm, procedure for finding the greatest common divisor (GCD) of two numbers, described by the Greek mathematician Euclid in his Elements (c.

bc). The method is computationally efficient and, with minor modifications, is still used by computers. The algorithm involves. Euclid's algorithm can be applied to real numbers, as described by Euclid in Book 10 of his Elements.

The goal of the algorithm is to identify a real number g such that two given real numbers, a and b, are integer multiples of it: a = mg and b = ng, where m and n are integers.[26].

Lets see the algorithm with an example now. Example of Subtraction Based Version: Lets find GCD of and Step 0: GCD of & Step 1: Applying Euclid’s algorithm the numbers now become & () Step 2: Applying again they become & 32() Step 3: And next – 32 & 84() Step 4: 32 & 52() Step 5: 32 &   Author of Edge lacing, Regular knots, An introduction to flat braids, Braiding, Generalizing Euclid's algorithm, via the regular and Moebius knot trees, order-n arithmetics, The braiding of row-coded regular knots, Braiding application, The regular knot.

Doing long division on paper with four digit denominators is not filed under "efficient" in my book. If I just have pen and paper available, I'd probably just guess that $\approx \cdot 23$, calculate the right-hand side, and work my way from there with the regular algorithm.

$\endgroup$ – Arthur Jan 1. Euclidean Algorithm on Brilliant, the largest community of math and science problem solvers. A new theory of braiding (RRI/2): algorithms for regular knots / A.G.

Schaake, J.C. Turner Generalizing Euclid's algorithm, via the regular and Moebius knot trees. Order-n arithmetics. Immediately download the Euclidean algorithm summary, chapter-by-chapter analysis, book notes, essays, quotes, character descriptions, lesson plans, and more - everything you need for studying or teaching Euclidean algorithm.

How to Find the GCF Using Euclid's Algorithm. Given two whole numbers where a is greater than b, do the division a ÷ b = c with remainder R. Replace a with b, replace b with R and repeat the division. Repeat step 2 until R=0. When R=0, the divisor, b, in the last equation is the greatest common factor, GCF.

Since greatest common factor (GCF. now be referred to as Euclid’s algorithm. Repeated operation by Q will Of course gcd (m19m2) = gcd Q(m17m2) and the algorithm ends when the first coordinate is zero. The second coordinate is then gcd (ml,m2).

Knuth states [ 11 that Euclid’s algorithm is the world’s oldest non-trivial algorithm. Join Raghavendra Dixit for an in-depth discussion in this video, Euclid's algorithm, part of Introduction to Data Structures & Algorithms in Java.

Euclid’s Algorithm. Euclid’s algorithm calculates the greatest common divisor of two positive integers a and b.

The algorithm rests on the obser-vation that a common divisor d of the integers a and b has to divide the diﬀerence a − b. Indeed, if a = a 0d and b = b0d for some integers a0 and b, then a−b = (a0 −b0)d; hence, d divides.

The Euclid’s algorithm (or Euclidean Algorithm) is a method for efficiently finding the greatest common divisor (GCD) of two numbers. The GCD of two integers X and Y is the largest number that divides both of X and Y (without leaving a remainder).

It is based on. The Euclidean algorithm appeared in Euclid's Elements, Book VII, Proposition 2 [37]; also see Book X, Propositions However, the idea is likely to have been known previously. Quoting Knuth again, "[w]e might call it the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day" [41, ].

$\begingroup$ The book that really helped me with this problem is "Introduction to algorithms" by Udi Manber. Chapter 2 "Mathematical induction" demonstrates many useful induction techniques with short and simple examples. $\endgroup$ – Panic Aug 27 '17 at that Euclid created this algorithm as a tool to continue his work on Number Theory.

The Euclidean Algorithm is discussed in propositions 1 and 2 in book VII. Proposition 1 states “When two unequal numbers are set out, and the less is continually subtracted in turn from the.

Basic Euclidean Algorithm for GCD The algorithm is based on below facts. If we subtract smaller number from larger (we reduce larger number), GCD doesn’t change.

So if we keep subtracting repeatedly the larger of two, we end up with GCD. Now instead of subtraction, if we divide smaller number, the algorithm stops when we find remainder 0. The Euclidean Algorithm. This is the currently selected item. Next lesson.

Primality test. Sort by: Top Voted. Modular inverses. Our mission is to provide a free, world-class education to anyone, anywhere. Khan Academy is a (c)(3) nonprofit organization. Donate or. Abstract. Perhaps the most famous equation in mathematics is A 2 + B 2 = C 2, which is known as Pythagoras’ problem of finding positive integer solutions for it is an ancient one indeed.

It is known to have been tackled successfully in the Babylonian period of more than years ago, for a clay tablet bearing a list of solutions in sexagesimal notation, dated at circa B.C. The Euclidean Algorithm is one of the oldest numerical algorithms still in use today.

Attributed to ancient Greek mathematician Euclid in his book “Elements” written approximately. What the algorithm looks like in words.

Euclid solved the problem graphically. He said If you have two distances, AB and CD, and you always take away the smaller from the bigger, you will end up with a distance that measures both of them.

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 is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c.

BC). It is an example of an algorithm, a step-by-step procedure for. Algorithm states GCD(a,b) = GCD(b,a%b). Consider a>b. We need to prove GCD(a,b) = GCD(b,a-b) first which can then be extended to modulo. Lets say g is the GCD(a,b).

To prove - GCD(b,a-b) is also g. If g is the greatest common factor of a and b - t. fractions, has been recognised in some knot theory books (for example, in Cromwell’s book on knots and links [2]).

However, in this paper, we will take it a step further by proving a special property of the Euclidean algorithm in Section 5 which will help us optimise the untangling algorithm. 1 arXivv2 [] 26 Mar The Euclidean algorithm is arguably one of the oldest and most widely known algorithms.

It is a method of computing the greatest common divisor (GCD) of two integers a a a and b b allows computers to do a variety of simple number-theoretic tasks, and also serves as a foundation for more complicated algorithms in number theory.

A graphical interpretation of Euclid's algorithm for calculating the greatest common divisor of two numbers: Given numbers and, draw a rectangle with width and this rectangle is divided into squares as shown in the Demonstration, then the width of the smallest square (shown in red) is the greatest common divisor of and.

The main application that comes to my mind is in implementation of a Rational number class. For example, the Python class Fraction uses the Euclidean Algorithm after every operation in order to simplify its fraction representation. Python has arbi. Step 3: Reset the algorithm to find the greatest common divisor of 6 and 4.

E.g. fit one 4x4 square in the remaining 4x6 unshaded rectangle. E.g. fit one 4x4 square in. Euclid Algorithm for Set of Integers: ‘Reduce’ vs. trees in R Posted on May 7, by msuzen in R bloggers | 0 Comments [This article was first published on Memo's Island, and kindly contributed to R-bloggers ].