Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Descartes-Frenicle-Sorli and Dris Conjectures Regarding Odd
  Perfect Numbers   Dris conjectured in his masters thesis that the inequality $q^k < n$ always
holds, if $N = {q^k}{n^2}$ is an odd perfect number with special prime $q$. In
this note, we initially show that either of the two conditions $n < q^k$ or
$\sigma(q)/n < \sigma(n)/q$ holds. This is achieved by first proving that
$\sigma(q)/n \neq \sigma(n)/q^k$, where $\sigma(x)$ is the sum of the divisors
of $x$. Using this analysis, we further show that the condition $q < n < q^k$
holds in four out of a total of six cases. Finally, we prove that $n < q^k$,
and that this holds unconditionally. This finding disproves both the Dris
Conjecture and the Descartes-Frenicle-Sorli Conjecture that $k = 1$.

20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
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The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat