Unique polynomial solution of $m/n=1/x+1/y+1/z$ for $n \equiv b {\rm mod}\, a$ if $(a,m)=1$

Abstract

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m/n=1/x(\lambda)+1/y(\lambda)+1/z(\lambda)$ with $n=b+a\lambda$ are explicitly given for a,b coprime and a not a multiple of m . The solution has the form $x(\lambda)=kn(\lambda)$, $y(\lambda)=n(\lambda)(s+r\lambda)$, $z(\lambda)=(kl/r)(s+r\lambda)$ where parameters $k,l,s,r\in \mathbb{Z}$ obey certain conditions depending on $a,b$. The conditions imply restrictions for some choices of $a,b$ which differ from the ones known in the case $m=4$. E.g., the modulus must be of the form $l(mk-1)$. One can also deduce that primes of the form $1+4K$ are excluded as modulus. Also if $a=p\ne m$ is prime and $b=a+1$, i.e., $n\equiv 1{\rm mod}\, p$, polynomial solutions are shown to be impossible. All results are valid for integers $m \ge 4$.