A note on the products $((m+1)^{2}+1)((m+2)^{2}+1)\hdots(n^{2}+1)$ and $((m+1)^{3}+1)((m+2)^{3}+1)\hdots(n^{3}+1)$

Abstract

We prove that for any positive integer $m$ there exists a positive real number $N_m$ such that whenever the integer $n\geq N_m$ neither the product $P^{n}_{m}=((m+1)^{2}+1)((m+2)^{2}+1)\hdots(n^{2}+1)$ nor the product $Q^{n}_{m}=((m+1)^{3}+1)((m+2)^{3}+1)\hdots(n^{3}+1)$ is a square.