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arXiv · 1312.6001

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

Keneth Adrian Precillas Dagal·Jose Arnaldo Bebita Dris
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Abstract

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 $\sigma(q)/n < \sigma(n)/q$ 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 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 $k = 1$.

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Keneth Adrian Precillas Dagal, Jose Arnaldo Bebita Dris. 2013-11-30. On the Descartes-Frenicle-Sorli and Dris Conjectures Regarding Odd Perfect Numbers. https://arxiv.org/abs/1312.6001

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