Some observations on the Diophantine equation $f(x)f(y)=f(z)^2$
Volume 142 / 2016
Colloquium Mathematicum 142 (2016), 275-283
MSC: Primary 11D72, 11D25; Secondary 11D41.
DOI: 10.4064/cm142-2-8
Abstract
Let $f\in \mathbb {Q}[X]$ be a polynomial without multiple roots and with $\mathop{\rm deg}(f)\geq 2$. We give conditions for $f(X)=AX^2+BX+C$ such that the Diophantine equation $f(x)f(y)=f(z)^2$ has infinitely many nontrivial integer solutions and prove that this equation has a rational parametric solution for infinitely many irreducible cubic polynomials. Moreover, we consider $f(x)f(y)=f(z)^2$ for quartic polynomials.
