members, the equation $p^x$ + $q^y$ - $r^z$ = 0 (where r is odd integer) has only positive integr solutions iff the following conditions made.
a) p = -1 (mod q^(2k)), here k is any positive integer.
b) 4|q -3
c) $p^2$ + q^(2k-1) = r
Then our equation $p^x$ + $q^y$ -$r^z$ = 0 has positive solutions in the form of (2, 2k-1, 1).
I am almost did. But, I need little more discussion to reach this solution and I want to know that if r is not odd, how to guess the solutions.
Thanks in advance to all members of stack exchange.