On the Products $(1^{\ell }+1)(2^{\ell }+1)\cdots (n^{\ell }+1)$, II

Abstract

In this paper, the following results are proved: (i) For any odd integer $\ell$ with at most two distinct prime factors and any positive integer $n$, the product $(1^{\ell }+1)(2^{\ell }+1)\cdots (n^{\ell }+1)$ is not a powerful number; (ii) For any integer $r\ge 1$, there exists a positive integer $T_r$ such that, if $\ell$ is a positive odd integer with at most $r$ distinct prime factors and $n$ is an integer with $n\ge T_r$, then $(1^{\ell }+1)(2^{\ell }+1)\cdots (n^{\ell }+1)$ is not a powerful number.

arXiv comment13pages