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

Abstract

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 note, we prove that (i) for any odd prime power $\ell$ and $n\ge \max \{q,11-q\}$, the product $(1^{\ell }+q^{\ell })(2^{\ell }+q^{\ell })\cdots (n^{\ell }+q^{\ell })$ is not a powerful number; (2) for any positive odd integer $\ell$, there exists an integer $N_{q,\ell }$ such that for any positive integer $n\ge N_{q,\ell }$, the product $(1^{\ell }+q^{\ell })(2^{\ell }+q^{\ell })\cdots (n^{\ell }+q^{\ell })$ is not a powerful number.

7 pages

References in corpus (1)

• Squares in Polynomial Product Sequences