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

Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 4, Pages 33–38
DOI: 10.7546/nntdm.2020.26.4.33-38
Full paper (PDF, 193 Kb)

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Authors and affiliations

Jose Arnaldo Bebita Dris
M. Sc. Graduate, Mathematics Department
De La Salle University, Manila, Philippines 1004

Immanuel Tobias San Diego
Department of Mathematics and Physical Sciences
Trinity University of Asia, Quezon City, Philippines 1102

Abstract

We 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 \big(m^2 - p^k \text{ is a power of two }\big) \land \big(p \text{ is a Fermat prime}\big). We end by stating some conjectures.

Keywords

  • Sum of divisors
  • Sum of aliquot divisors
  • Deficiency
  • Odd perfect number
  • Special prime

2010 Mathematics Subject Classification

  • 11A05
  • 11A25

References

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Cite this paper

Dris, J. A. B., & San Diego, I. T. (2020). On the quantity m2 − pk where pkm2 is an odd perfect number, 26(4), 33-38, DOI: 10.7546/nntdm.2020.26.4.33-38.

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