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

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 $n\ge N_l$, ${\prod }_{k=1}^n(2k^2+l)$ 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 $l$.