Products of consecutive values of some quartic polynomials

Abstract

In this paper, we investigate some special quartic polynomials $P$ whose coefficients for $x^4,x^3,\dots,1$ are $a^2, 2a(a+b), a^2+b^2+3ab+2ac, (a+b)(b+2c), (a+b+c)c$, where $a,b,c \in \mathbb{Z}$, and consider the question whether the product $\prod_{k=1}^m P(k)$ is a perfect square for infinitely many $m \in \mathbb{N}$ or for only finitely many $m \in \mathbb{N}$. The answer depends on the solutions of the Pell type diophantine equation $(a+b+c)(ax^2+bx+c)=y^2$. Our results imply, for example, that the product $\prod_{k=1}^m (4k^4+8k^2+9)$ is a perfect square for infinitely many $m \in \mathbb{N}$, whereas the product $\prod_{k=1}^m (k^4+7k^2+16)$ is a perfect square for $m=3$ only, when it equals $230400=480^2$.