An infinite collection of quartic polynomials whose products of consecutive values are not perfect squares

Abstract

Using an elementary identity, we prove that for infinitely many polynomials $P(x)\in \mathbb{Z}[X]$ of fourth degree, the equation $\prod\limits_{k=1}^{n}P(k)=y^2$ has finitely many solutions in $\mathbb{Z}$. We also give an example of a quartic polynomial for which the product of it's first consecutive values is infinitely often a perfect square.