On the total number of prime factors of an odd perfect number

Abstract:

We say $n\in {\mathbb N}$ is perfect if $\sigma (n)=2n$, where $\sigma (n)$ denotes the sum of the positive divisors of $n$. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization 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 $p\equiv \alpha \equiv 1\pmod 4$. We prove that if $\beta _j\equiv 1\pmod 3$ or $\beta _j\equiv 2\pmod 5$ for all $j$, $1\le j\le k$, then $3\nmid n$. We also prove as our main result that $\Omega (n)\ge 37$, where $\Omega (n)=\alpha +2\sum _{j=1}^{k}\beta _j$. This improves a result of Sayers $( \Omega (n)\ge 29 )$ given in 1986.