Does anyone know how to find integer solutions of the quadratic equation
where $z$ is a fixed odd prime or $1$ and $f$ is a fixed odd prime greater than $3$?
This problem arose from the Diophantine equation $A+B=C$ where $A,B,C$ are natural numbers with no common factor. The managers of this site asked me to make my questions harder for this reason I will restate the above. Does anyone know if the quadratic equation $x^2-2x-[a^5+b^5]=0$ has infinite integer solutions?