Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Products $(1^\ell+1)(2^\ell+1)\cdots (n^\ell +1)$, II   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.

12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ