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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


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.

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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

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