Powerful numbers in $(1^{\ell }+q^{\ell })(2^{\ell }+q^{\ell })\cdots (n^{\ell }+q^{\ell })$

Quan-Hui Yang, Qing-Qing Zhao

Let $q$ be a positive integer. Recently, Niu and Liu proved that if $n\ge \max \{q,1198-q\}$, then the product $(1^3+q^3)(2^3+q^3)\cdots (n^3+q^3)$ is not a powerful number. In this…