On the Descartes-Frenicle-Sorli and Dris Conjectures Regarding Odd Perfect Numbers

Dris conjectured in his masters thesis that the inequality $q^k<n$ always holds, if $N=q^k{n}^2$ is an odd perfect number with special prime $q$. In this note, we initially show that either of the two conditions $n<q^k$ or $\sigma (q)\mathrm{/}n<\sigma (n)\mathrm{/}q$ holds. This is achieved by first proving that $\sigma (q)\mathrm{/}n\mathrm{\ne }\sigma (n)\mathrm{/}{q}^k$, where $\sigma (x)$ is the sum of the divisors of $x$. Using this analysis, we further show that the condition $q<n<q^k$ holds in four out of a total of six cases. Finally, we prove that $n<q^k$, and that this holds unconditionally. This finding disproves both the Dris Conjecture and the Descartes-Frenicle-Sorli Conjecture that $k=1$.