Upper bounds on the second largest prime factor of an odd perfect number

Abstract

Acquaah and Konyagin showed that if $N$ is an odd perfect number where $N=p_1^{a_1}{p}_2^{a_2}\cdots p_k^{a_k}$ where $p_1<p_2\cdots <p_k$ then one must have $p_k<3^{1/3}{N}^{1/3}$. Using methods similar to theirs, we show that $p_{k-1}<(2N{)}^{1/5}$ and that $p_{k-1}{p}_k<6^{1/4}{N}^{1/2}.$ We also show that if $p_k$ and $p_{k-1}$ are close to each other than these bounds can be further strengthened.

11 pages, Accepted to International Journal of Number Theory