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We prove that m − p is not a square, if n = pm is an odd perfect number with special prime p, under the hypothesis that σ(m)/p 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 ( m − p is a power of two ) ∧ ( p is a Fermat prime ) . We end by stating some conjectures.

On the quantity m^2 − p^k where p^k m^2 is an odd perfect number