A note on the squares of the form $\prod_{k=1}^n (2k^2+l)$ with $l$ odd

TL;DR

This paper investigates when the product of quadratic forms involving an odd integer $l$ results in perfect squares, establishing bounds and identifying specific cases where the product is a square.

Contribution

It provides an explicit lower bound for $n$ beyond which the product cannot be a square, and determines all such $n$ for certain odd $l$ values using Cilleruelo's method.

Findings

For large $n$, the product is not a perfect square.

Explicit bounds depend on the odd integer $l$.

Certain small $n$ values yield perfect squares for specific $l$.

Abstract

Let $l$ be a positive odd integer. Using Cilleruelo's method, we establish an explicit lower bound $N_l$ depending on $l$ such that for all $n\geq N_l$, $\prod_{k=1}^n (2k^2+l)$ is not a square. As an application, we determine all values of $n$ such that $\prod_{k=1}^n (2k^2+l)$ is a square for certain values of $l$.