New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number

Abstract

Let $\sigma (n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $\sigma (n)=2n$. 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 $p\equiv \alpha \equiv 1(\mathrm{mod}4)$. Define the total number of prime factors of $N$ as $\Omega (N):=\alpha +2{\sum }_{j=1}^k{\beta }_j$. Sayers showed that $\Omega (N)\ge 29$. This was later extended by Iannucci and Sorli to show that $\Omega (N)\ge 37$. This was extended by the author to show that $\Omega (N)\ge 47$. Using an idea of Carl Pomerance this paper extends these results. The current new bound is $\Omega (N)\ge 75$.

9 pages