Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New techniques for bounds on the total number of Prime Factors of an Odd
  Perfect Number   Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We say that
$n$ is perfect if $\sigma(n) = 2 n$. Currently there are no known odd perfect
numbers. It is known that if an odd perfect number exists, then it must be of
the form $N = p^\alpha \prod_{j=1}^k q_j^{2 \beta_j}$, where $p, q_1, ..., q_k$
are distinct primes and $p \equiv \alpha\equiv 1 \pmod{4}$. Define the total
number of prime factors of $N$ as $\Omega(N) := \alpha + 2 \sum_{j=1}^k
\beta_j$. Sayers showed that $\Omega(N) \geq 29$. This was later extended by
Iannucci and Sorli to show that $\Omega(N) \geq 37$. This was extended by the
author to show that $\Omega(N) \geq 47$. Using an idea of Carl Pomerance this
paper extends these results. The current new bound is $\Omega(N) \geq 75$.

0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
19
The categories of this paper are: eess