## Taiwanese Journal of Mathematics

### THE RESTRICTED SOLUTIONS OF $ax+by=\gcd(a,b)$

Ju-Si Lee

#### Abstract

Let $D$ denote a principle ideal domain with identity element $1$. Fix three elements $a, b, d$ in $D$ with ${\rm gcd}(a, b)=d$. We show there exist two elements $x, y$ in $D$ with either ${\rm gcd}(a, y)=1$ or ${\rm gcd}(b, x)=1$ such that $ax+by=d$. Moreover we show there exist $x, y$ in $D$ such that ${\rm gcd}(a, y)=1,$ ${\rm gcd}(b, x)=1$ and $ax+by=d$ if and only if for each prime divisor $p$ of $d$ with a complete set of residues modulo $p$ containing exactly $2$ elements, the power of $p$ appearing in the factorization of $a$ is different to that of $b$. We apply our results to the study of double-loop networks..

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 5 (2008), 1191-1199.

Dates
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.twjm/1500574257

Digital Object Identifier
doi:10.11650/twjm/1500574257

Mathematical Reviews number (MathSciNet)
MR2431889

Zentralblatt MATH identifier
1195.11008

#### Citation

Lee, Ju-Si. THE RESTRICTED SOLUTIONS OF $ax+by=\gcd(a,b)$. Taiwanese J. Math. 12 (2008), no. 5, 1191--1199. doi:10.11650/twjm/1500574257. https://projecteuclid.org/euclid.twjm/1500574257

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