A note on the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^{\alpha}$ and application to odd perfect numbers

TL;DR

This paper investigates properties of odd perfect numbers, showing that certain ratios involving their factors are restricted unless a specific exponent equals one, using advanced number theory and Diophantine equations.

Contribution

It establishes new restrictions on the structure of odd perfect numbers by analyzing a specific Diophantine equation and prime ideal factorizations, extending to multiply perfect numbers.

Findings

The ratio σ(n^2)/q^α is not a square or prime times a square unless α=1.

The proof involves properties of quadratic orders and primitive solutions of generalized Fermat equations.

Provides a generalization to odd multiply perfect numbers.

Abstract

Let $N$ be an odd perfect number. Then, Euler proved that there exist some integers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmid n$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that the ratio $\frac{\sigma(n^{2})}{q^{\alpha}}$ is neither a square nor a square times a single prime unless $\alpha = 1$. It is a direct consequence of a certain property of the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^{\alpha}$, where $l$ denotes one or a prime, whose proof is based on the prime ideal factorization in the quadratic orders $\mathbb{Z}[\sqrt{1-q}]$ and the primitive solutions of generalized Fermat equations $x^{\beta}+y^{\beta} = 2z^{2}$. We give also a slight generalization to odd multiply perfect numbers.