# Solving the Odd Perfect Number Problem: Some Old and New Approaches

@article{Dris2012SolvingTO, title={Solving the Odd Perfect Number Problem: Some Old and New Approaches}, author={Jose Arnaldo B. Dris}, journal={arXiv: Number Theory}, year={2012} }

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems of number theory. This thesis presents some of the old as well as new approaches to solving the OPN Problem. In particular, a conjecture predicting an injective and surjective mapping $X = \sigma(p^k)/p^k, Y = \sigma(m^2)/m^2$ between OPNs $N = {p^k}{m^2…

## 24 Citations

### Solving the Odd Perfect Number Problem: Some New Approaches

- Mathematics
- 2012

A conjecture predicting an injective and surjective mapping $X = \displaystyle\frac{\sigma(p^k)}{p^k}, Y = \displaystyle\frac{\sigma(m^2)}{m^2}$ between OPNs $N = {p^k}{m^2}$ (with Euler factor…

### A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers

- Mathematics
- 2013

Let $\sigma(x)$ be the sum of the divisors of $x$. If $N$ is odd and $\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with…

### Euclid-Euler Heuristics for Perfect Numbers

- Mathematics
- 2013

An odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. Similarly, an even perfect number $M$ is…

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

- Mathematics
- 2013

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…

### On Dris Conjecture about Odd Perfect Numbers

- Mathematics
- 2017

The Euler's form of odd perfect numbers, if any, is $n=\pi^{\alpha}N^2$, where $\pi$ is prime, $(\pi,N)=1$ and $\pi\equiv \alpha \equiv 1 \pmod{4}$. Dris conjecture states that $N>\pi^{\alpha}$. We…

### A Partial Proof of a Conjecture of Dris

- Mathematics
- 2016

Euler showed that if an odd perfect number $N$ exists, it must consist of two parts $N=q^k n^2$, with $q$ prime, $q \equiv k \equiv 1 \pmod{4}$, and gcd$(q,n)=1$. Dris conjectured that $q^k < n$. We…

### A note on odd perfect numbers

- MathematicsArXiv
- 2011

It is shown that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it, then $\sigma(N/q^{alpha})/q^(\alpha}>5$ is bounded by some function depending on $K$.

### New Results for Sorli's Conjecture on Odd Perfect Numbers

- Mathematics
- 2013

If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then Sorli's conjecture predicts that $k=1$. In this article, we give a strategy for trying to prove that the inequality $n < q$ is…

### New Results for the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers

- Mathematics
- 2013

If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then the Descartes-Frenicle-Sorli conjecture predicts that $k=1$. Brown has recently announced a proof for the inequality $q < n$,…

### 1 N ov 2 01 3 The Abundancy Index of Divisors of Odd Perfect Numbers-Part II ✩ Keneth

- Mathematics
- 2014

A positive integer M is said to be almost perfect if σ(M) = 2M − 1, where σ(x) is the sum of the divisors of x. If N is odd and σ(N) = 2N , then the odd perfect number N is said to be given in…

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