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On a Conjecture of Dris Regarding Odd Perfect Numbers

Dris conjectured (in his M.Sc. thesis) that the inequality $q^k < n$ always holds, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form. In this short note, we show that either of the two conditions $n < q^k$ or $\sigma(q)/n < \sigma(n)/q$ holds. This is achieved by first proving that $\sigma(q)/n \neq \sigma(n)/q^k$, where $\sigma(x)$ is the sum of the divisors of $x$. Using this analysis, we show that the inequalities $q < n < q^k$ hold in four out of a total of six cases. By utilizing a separate analysis, we show that the condition $n < q < n\sqrt{3}$ holds in four out of a total of five cases. We conclude with some open problems related to Sorli's conjecture that $k = 1$.