Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: More on the Nonexistence of Odd Perfect Numbers of a Certain Form   Euler showed that if an odd perfect number exists, it must be of the form $N
= p^\alpha q_{1}^{2\beta_{1}}$ $\ldots$ $q_{k}^{2\beta_{k}}$, where $p, q_{1},
\ldots, q_k$ are distinct odd primes, $\alpha$, $\beta_{i} \geq 1$, for $1 \leq
i \leq k$, with $p \equiv \alpha \equiv 1 \pmod{4}$. In 2005, Evans and
Pearlman showed that $N$ is not perfect, if $3|N$ or $7|N$ and each $\beta_{i}
\equiv 2 \pmod{5}$. We improve on this result by removing the hypothesis that
$3|N$ or $7|N$ and show that $N$ is not perfect, simply, if each $\beta_{i}
\equiv 2 \pmod{5}$.

35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics