# Diophantine Equation: $f(x)f(y) = f(z^2)$ where $f$ is quadratic

In the study of the Diophantine Equation $f(x)f(y) = f(z^2)$ where $f$ is quadratic, the computational proofs I have seen (for specific $f$) rely on Pell's Equation.

For example, if $f(t) = t^2+t+1$, then the equation becomes

$$(x^2+x+1)(y^2+y+1) = (z^4+z^2+1)$$

and via the substitution $y=x+2z$, and one more subsequent substitution, one eventually reaches the generalised Pell Equation

$$X^2-2Z^2=-7$$

which is known to have infinitely many positive integer solutions.

Has anyone come across other methods of showing infinitely many (nontrivial) integer or rational solutions for quadratic $f$ that do not rely on Pell Equations?

Thank you.

• You might try $x^2 + x - 1,$ which has positive discriminant. Meanwhile, where does this come from? How is it that you are searching for such relations? – Will Jagy Jan 25 '13 at 2:13
• Thanks Will, I'm studying this particular equation at the moment, but have not seen much written about it besides reducing to Pell. – Conan Wong Jan 25 '13 at 3:04