Download PDFOpen PDF in browserOn the Quantity $m^2 - p^k$ Where $p^k m^2$ is an Odd Perfect NumberEasyChair Preprint 39677 pages•Date: July 29, 2020AbstractWe prove that $m^2 - p^k$ is not a square, if $n = p^k m^2$ is an odd perfect number with special prime $p$, under the hypothesis that $\sigma(m^2)/p^k$ 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 $$\bigg(m^2 - p^k \text{ is a power of two }\bigg) \land \bigg(p \text{ is a Fermat prime}\bigg).$$ We end by stating some conjectures. Keyphrases: Deficiency, Odd perfect number, Special prime, Sum of aliquot divisors, Sum of divisors
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