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

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..