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

TL;DR

This paper constructs an infinite family of quartic polynomials with the property that the product of their first n values is rarely a perfect square, and provides an example where this product is often a perfect square.

Contribution

It proves the existence of infinitely many quartic polynomials with finitely many perfect square products of consecutive values, and presents an example with infinitely many perfect square products.

Findings

Infinitely many quartic polynomials have finitely many perfect square products of initial values.

An explicit quartic polynomial has infinitely many perfect square products of its initial values.

Elementary identities can be used to analyze perfect square products in polynomial sequences.

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.