A note on the Diophantine equations 2ln2=1+q+⋯+qα 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 $q\equiv \alpha \equiv 1mod4$. In this note, we prove that the ratio $\frac{\sigma \left(n^2\right)}{q^{\alpha }}$ 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 $2l{n}^2=1+q+\cdots +q^{\alpha }$, where $l$ denotes one or a prime, and its proof is based on the prime ideal factorization in the quadratic orders $Z\left[\sqrt{1-q}\right]$ 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.