# On Dris Conjecture Regarding Odd Perfect Numbers

OpenYear of origin: 2008

Posted online: 2019-03-16 07:30:46Z by Jose Arnaldo Bebita Dris47

Cite as: P-190316.1

• Number Theory
• General Mathematics

### Problem's Description

If $N = q^k n^2$ is an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$, then it is somewhat trivial to prove that $$I(q^k) < \frac{5}{4} < \sqrt{\frac{8}{5}} < I(n),$$ which led Dris to conjecture that $q^k < n$. Brown, Starni, and Dris has shown that $q^k < n$ follows from the truth of the Descartes-Frenicle-Sorli Conjecture that $k=1$. In fact, it is known (by work of Dris) that $$k = 1 \implies \sigma(q^k) < n,$$ where $\sigma(x)$ is the sum of divisors of the positive integer $x$. The problem is to prove $q^k < n$ unconditionally. (In other words, it would suffice to prove the implication $k > 1 \implies q^k < n$.)

3. ## Article On Dris conjecture about odd perfect numbers

Notes on Number Theory and Discrete Mathematics 24 (1), 5-9, 2018arXivfulltext

4. ## ThesisIs an originSolving the Odd Perfect Number Problem: Some Old and New Approaches

pp. 134, year of publication: 2008arXivfulltext