On the third largest prime divisor of an odd perfect number

Abstract

Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as the closely related problem of improving bounds $bc$, and $abc$. In particular, we prove two results. First we prove a new general bound on any prime divisor of an odd perfect number and obtain as a corollary of that bound that $$a<2N^{\frac{1}{6}}.$$ Second, we show that $$abc<(2N{)}^{\frac{3}{5}}.$$ We also show how in certain circumstances these bounds and related inequalities can be tightened. Define a ${\sigma }_{m,n}$ pair to be a pair primes $p$ and $q$ where $q|\sigma (p^m)$, and $p|\sigma (q^n)$. Many of our results revolve around understanding ${\sigma }_{2,2}$ pairs. We also prove results concerning ${\sigma }_{m,n}$ pairs for other values of $m$ and $n$.

References in corpus (2)

• A congruence involving harmonic sums modulo $p^{\alpha }{q}^{\beta }$
• An improvement of an inequality of Ochem and Rao concerning odd perfect numbers