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