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Jose Arnaldo Bebita Dris

Publications and source records attributed to Jose Arnaldo Bebita Dris.

6 recordsLinked to original sources
arXiv2022

On the quantity $m^2-p^k$ where $p^k m^2$ is an odd perfect number -- Part II

Let $p^k m^2$ be an odd perfect number with special prime $p$. Extending previous work of the authors, we prove that the inequality $m < p^k$ follows from $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$, under the following hypotheses: (a) $m > t > 2^r$, or (b) $m > 2^r > t$. We also prove that the estimate $m^2 - p^k > 2m$ holds. We can also improve this unconditional estimate to $m^2 - p^k > {313m^2}/315$.

math.NT↗
arXiv2022

A new approach to odd perfect numbers via GCDs

Let $q^k n^2$ be an odd perfect number with special prime $q$. Define the GCDs $$G = \gcd\bigg(σ(q^k),σ(n^2)\bigg)$$ $$H = \gcd\bigg(n^2,σ(n^2)\bigg)$$ and $$I = \gcd\bigg(n,σ(n^2)\bigg).$$ We prove that $G \times H = I^2$. (Note that it is trivial to show that $G \mid I$ and $I \mid H$ both hold.) We then compute expressions for $G, H,$ and $I$ in terms of $σ(q^k)/2, n,$ and $\gcd\bigg(σ(q^k)/2,n\bigg)$. Afterwards, we prove that if $G = H = I$, then $σ(q^k)/2$ is not squarefree. Other natural and related results are derived further. Lastly, we conjecture that the set $$\mathscr{A} = \{m : \gcd(m,σ(m^2))=\gcd(m^2,σ(m^2))\}$$ has asymptotic density zero.

math.NT↗
arXiv2021

On the quantity $I(q^k) + I(n^2)$ where $q^k n^2$ is an odd perfect number -- Part II

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 $k$.

math.GM↗
arXiv2020

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 $σ(q)/n < σ(n)/q$ 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 $k = 1$.

math.NT↗
arXiv2020

Some Modular Considerations Regarding Odd Perfect Numbers

Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $σ(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application of this result to the case when $σ(m^2)/p^k$ is a square.

math.NT↗