Does anyone know how to find integer solutions of the quadratic equation
$$y^2+y+z=f$$
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?
$$ y^2 + y + z = f $$ $$ y^2 + y + z - f = 0 $$ $$ ay^2 + by + c = 0 $$ where $a=1$, $b=1$, and $c=z-f$. So $$ y = \frac{-b\pm\sqrt{b^2-4ac}}{2a} = \frac{-1\pm\sqrt{1-4(z-f)}}{2} $$ In its method of solution, it's no different from any other quadratic equation.
Whether the solutions are integers depends of course on $z$ and $f$.