On a theorem of Kanold on odd perfect numbers

Tomohiro Yamada

doi:10.1007/s13226-023-00530-y

paper

On a theorem of Kanold on odd perfect numbers

arXiv:2312.17059 · doi:10.1007/s13226-023-00530-y

Abstract

We shall prove that if $N = p^{α} q_{1}^{2 β_{1}} q_{2}^{2 β_{2}} \dots q_{r - 1}^{2 β_{r - 1}}$ is an odd perfect number such that $p, q_{1}, \dots, q_{r - 1}$ are distinct primes, $p \equiv α \equiv 1 mod 4$ and $t$ divides $2 β_{i} + 1$ for all $i = 1, 2, \dots, r - 1$ , then $t^{5}$ divides $N$ , improving an eighty-year old result of Kanold.

arXiv comment8 pages, the author's final version of the paper of the same title in Indian J. Pure Appl. Math, which is available at https://doi.org/10.1007/s13226-023-00530-y