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 \(m \ge 4\).