A note on the Diophantine equations 2ln²=1+q+⋯+q^α and application to odd perfect numbers

Let N be an odd perfect number. Then, Euler proved that there exist some integers n,α and a prime q such that N=n²q^α, q∤n, and q≡α≡1mod4. In this note, we prove that the ratio [Formula presented] is neither a square nor a square times a single prime unless α=1. It is a direct consequence of a certain property of the Diophantine equation 2ln²=1+q+⋯+q^α, where l denotes one or a prime, and its proof is based on the prime ideal factorization in the quadratic orders Z[1−q] and the primitive solutions of generalized Fermat equations x^β+y^β=2z². We give also a slight generalization to odd multiply perfect numbers.