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

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation m/n=1/x () +1/y () +1/z () with n=b+a are explicitly given for a, b coprime and a not a multiple of m. The solution has the form x () =kn (), y () =n () (s+r), z () = (kl/r) (s+r) where parameters k, l, s, r 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 m is prime and b=a+1, i. e. , n 1 mod\, p, polynomial solutions are shown to be impossible. All results are valid for integers m 4.