# On Dris Conjecture about Odd Perfect Numbers

## TL;DR

This paper investigates Dris's conjecture on odd perfect numbers, deriving new inequalities involving their prime factorization and providing bounds on their components.

## Contribution

It establishes a lower bound for N^2 in terms of π and α, advancing understanding of the structure of hypothetical odd perfect numbers.

## Key findings

- N^2 > (1/2)π^γ, with γ=max{ω(n)-1, α}
- ω(n) ≥ 9, the number of distinct prime factors of n
- Provides new inequalities related to the Euler form of odd perfect numbers

## Abstract

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 find that $N^2>\frac{1}{2}\pi^{\gamma}$, with $\gamma=max\{\omega(n)-1,\alpha\}$; $\omega(n)\geq 9$ is the number of distinct prime factors of $n$.