The Euclidean Algorithm as a Matrix Transformation

1972 ◽  
Vol 65 (3) ◽  
pp. 228-229
Author(s):  
Aziz Ibrahim ◽  
Edward Gucker
Keyword(s):  
Euclidean Algorithm ◽  
Matrix Transformation ◽  
Greatest Common Divisor ◽  
Repeated Application ◽  
Division Algorithm ◽  
Positive Integers

The algorithm of Euclid for finding the greatest common divisor of two positive integers is based on repeated application of the division algorithm.

Microcomputer-assisted Discoveries: Euclidean Algorithm and Continued Fractions

1983 ◽  
Vol 76 (7) ◽  
pp. 510-548
Author(s):  
Clark Kimberling
Keyword(s):  
Continued Fractions ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Positive Integers

Students can use microcomputers to cut through algorithms and computations to gain mathematical insights. This approach is especially true for the Euclidean algorithm, so often used to find the greatest common divisor (GCD) of two positive integers. The Euclidean algorithm also yields continued fractions, at least far enough for students to find patterns and discover truths about numbers.

1. Numbers and algebra

2015 ◽  
pp. 1-10
Author(s):  
Peter M. Higgins
Keyword(s):  
Fundamental Theorem ◽  
Common Factor ◽  
Number System ◽  
Rational Numbers ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Natural Numbers ◽  
Division Algorithm ◽  
Fundamental Theorem Of Arithmetic ◽  
And Algebra

‘Numbers and algebra’ introduces the number system and explains several terms used in algebra, including natural numbers, positive and negative integers, rational numbers, number factorization, the Fundamental Theorem of Arithmetic, Euclid’s Lemma, the Division Algorithm, and the Euclidean Algorithm. It proves that any common factor c of a and b is also a factor of any number of the form ax + by, and since the greatest common divisor (gcd) of a and b has this form, which may be found by reversing the steps of the Euclidean Algorithm, it follows that any common factor c of a and b divides their gcd d.

scholarly journals Another generalisation of smith's determinant

1989 ◽  
Vol 40 (3) ◽  
pp. 413-415 ◽  
Author(s):  
Scott Beslin ◽  
Steve Ligh
Keyword(s):  
Greatest Common Divisor ◽  
Euler’S Totient Function ◽  
The Matrix ◽  
Positive Integers

Let S = {x1, x2, …, xn} be a set of distinct positive integers. The n × n matrix [S] = (Sij), where Sij, = (xi, xj), the greatest common divisor of xi, and xj, is called the greatest common divisor (GCD) matrix on S. H.J.S. Smith showed that the determinant of the matrix [E(n)], E(n) = { 1,2, …, n}, is ø(1)ø(2) … ø(n), where ø(x) is Euler's totient function. We extend Smith's result by considering sets S = {x1, x2, … xn} with the property that for all i and j, (xi, xj) is in S.

MORE ON A CERTAIN ARITHMETICAL DETERMINANT

2017 ◽  
Vol 97 (1) ◽  
pp. 15-25 ◽  
Author(s):  
ZONGBING LIN ◽  
SIAO HONG
Keyword(s):  
Linear Algebra ◽  
Acta Math ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Mobius Function ◽  
Dirichlet Convolution ◽  
Positive Integers

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

scholarly journals On the number of arithmetic formulas

2015 ◽  
Vol 11 (04) ◽  
pp. 1099-1106 ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
The Other ◽  
Repeated Application ◽  
Positive Integers ◽  
Arithmetic Expressions

For each positive integers n, let g(n) be the number of arithmetic expressions evaluating to n and involving only the constant 1, additions and multiplications, with the restriction that multiplication by 1 is not allowed. We consider two arithmetic expressions to be equal if one can be obtained from the other through a repeated application of the commutative and associative properties. We give an algorithm to compute g(n) and prove that [Formula: see text], as n → +∞, where β ≔ log(24)/24.

scholarly journals A way to compute a greatest common divisor in the Galois field (GF (2^n ))

2019 ◽  
Vol 16 ◽  
pp. 8317-8321
Author(s):  
Waleed Eltayeb Ahmed
Keyword(s):  
Galois Field ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Multiplicative Inverse

This paper presents how the steps that used to determine a multiplicative inverse by method based on the Euclidean algorithm, can be used to find a greatest common divisor for polynomials in the Galois field (2^n ).

scholarly journals GCD of Aunu Binary Polynomials of Cardinality Seven Using Extended Euclidean Algorithm

2021 ◽  
Vol 09 (09) ◽  
Author(s):  
Ibrahim A. A. ◽  
Keyword(s):  
Finite Fields ◽  
Coding Theory ◽  
Error Correcting Codes ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Algebraic Structures ◽  
Extended Euclidean Algorithm ◽  
Encryption Schemes ◽  
Theory Error

Finite fields is considered to be the most widely used algebraic structures today due to its applications in cryptography, coding theory, error correcting codes among others. This paper reports the use of extended Euclidean algorithm in computing the greatest common divisor (gcd) of Aunu binary polynomials of cardinality seven. Each class of the polynomial is permuted into pairs until all the succeeding classes are exhausted. The findings of this research reveals that the gcd of most of the pairs of the permuted classes are relatively prime. This results can be used further in constructing some cryptographic architectures that could be used in design of strong encryption schemes.

scholarly journals Cuntz-Krieger Algebras Associated with Hilbert $C^*$-Quad Modules of Commuting Matrices

2015 ◽  
Vol 117 (1) ◽  
pp. 126 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Euclidean Algorithm ◽  
C Algebra ◽  
Commuting Matrices ◽  
Tiling Space ◽  
Positive Integers

Let $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ be the $C^*$-algebra associated with the Hilbert $C^*$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^\kappa$ is transitive, the $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ is simple and purely infinite. In particular, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{[N],[M]}_{\kappa}}$ are computed by using the Euclidean algorithm.

scholarly journals Lattice Paths Between Diagonal Boundaries

10.37236/1368 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Heinrich Niederhausen
Keyword(s):  
Boundary Conditions ◽  
Reflection Principle ◽  
Exact Distribution ◽  
Lattice Paths ◽  
Repeated Application ◽  
North East ◽  
Diagonal Band ◽  
Sample Statistic ◽  
Kolmogorov Smirnov ◽  
Positive Integers

A bivariate symmetric backwards recursion is of the form $d[m,n]=w_{0}(d[m-1,n]+d[m,n-1])+\omega_{1}(d[m-r_{1},n-s_{1}]+d[m-s_{1},n-r_{1}])+\dots+\omega_{k}(d[m-r_{k},n-s_{k}]+d[m-s_{k},n-r_{k}])$ where $\omega_{0},\dots\omega_{k}$ are weights, $r_{1},\dots r_{k}$ and $s_{1},\dots s_{k}$ are positive integers. We prove three theorems about solving symmetric backwards recursions restricted to the diagonal band $x+u < y < x-l$. With a solution we mean a formula that expresses $d[m,n]$ as a sum of differences of recursions without the band restriction. Depending on the application, the boundary conditions can take different forms. The three theorems solve the following cases: $d[x+u,x]=0$ for all $x\geq0$, and $d[x-l,x]=0$ for all $x\geq l$ (applies to the exact distribution of the Kolmogorov-Smirnov two-sample statistic), $d[x+u,x]=0$ for all $x\geq0$, and $d[x-l+1,x]=d[x-l+1,x-1]$ for $x\geq l$ (ordinary lattice paths with weighted left turns), and $d[y,y-u+1]=d[y-1,y-u+1]$ for all $y\geq u$ and $d[x-l+1,x]=d[x-l+1,x-1]$ for $x\geq l$. The first theorem is a general form of what is commonly known as repeated application of the Reflection Principle. The second and third theorem are new; we apply them to lattice paths which in addition to the usual North and East steps also make two hook moves, East-North-North and North-East-East. Hook moves differ from knight moves (covered by the first theorem) by being blocked by any piece of the barrier they encounter along their three part move.

Sign in / Sign up

Export Citation Format

Share Document