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

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m\mathrm{/}n=1\mathrm{/}x(\lambda )+1\mathrm{/}y(\lambda )+1\mathrm{/}z(\lambda )$ with $n=b+a\lambda$ are explicitly given for a,b coprime and a not a multiple of m . The solution has the form $x(\lambda )=kn(\lambda )$, $y(\lambda )=n(\lambda )(s+r\lambda )$, $z(\lambda )=(kl\mathrm{/}r)(s+r\lambda )$ 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 $a=p\mathrm{\ne }m$ is prime and $b=a+1$, i.e., $n\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}p$, polynomial solutions are shown to be impossible. All results are valid for integers $m\ge 4$.