On a theorem of Kanold on odd perfect numbers

Abstract

We shall prove that if $N=p^\alpha q_1^{2\beta_1} q_2^{2\beta_2} \cdots q_{r-1}^{2\beta_{r-1}}$ is an odd perfect number such that $p, q_1, \ldots, q_{r-1}$ are distinct primes, $p\equiv\alpha\equiv 1\mod{4}$ and $t$ divides $2\beta_i+1$ for all $i=1, 2, \ldots, r-1$, then $t^5$ divides $N$, improving an eighty-year old result of Kanold.