On the quantity m2 - pk where pkm2 is an odd perfect number

We prove that m2 - pk is not a square, if n = pkm2 is an odd perfect number with special prime p, under the hypothesis that sigma(m2)/pk is a square. We are also able to prove the same assertion without this hypothesis. We also show that this hypothesis is incompatible with the set of assumptions (m2 - pk is a power of two) Lambda (p is a Fermat prime). We end by stating some conjectures.