arXiv subjects
Publications and source records attributed to Keneth Adrian Precillas Dagal.
In this note, we continue an approach pursued in an earlier paper of the second author and thereby attempt to produce an improved lower bound for the sum $I(q^k) + I(n^2)$, where $q^k n^2$ is an odd perfect number with special prime $q$ and $I(x)$ is the abundancy index of the positive integer $x$. In particular, this yields an upper bound for $k$.
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 $σ(q)/n < σ(n)/q$ holds. This is achieved by first proving that $σ(q)/n \neq σ(n)/q^k$, where $σ(x)$ 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 $k = 1$.