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Some Modular Considerations Regarding Odd Perfect Numbers - Part II

EasyChair Preprint 2820

12 pagesDate: February 29, 2020

Abstract

In this article, we consider the various possibilities for p and k modulo 16, and show conditions under which the respective congruence classes for σ(m2) (modulo 8) are attained, if pk m2 is an odd perfect number with special prime p. We prove that

  1. σ(m2) ≡ 1 mod 8 holds only if p+k ≡ 2 mod {16}.
  2. σ(m2) ≡ 3 mod 8 holds only if p-k ≡ 4 mod {16}.
  3. σ(m2) ≡ 5 mod 8 holds only if p+k ≡ 10 mod {16}.
  4. σ(m2) ≡ 7 mod 8 holds only if p-k ≡ 4 mod {16}.

We express gcd(m2,σ(m2)) as a linear combination of m2 and σ(m2). We also consider some applications under the assumption that σ(m2)/pk is a square. Lastly, we prove a last-minute conjecture under this hypothesis.

Keyphrases: Deficiency, Odd perfect number, Special prime, Sum of aliquot divisors, Sum of divisors

BibTeX entry
BibTeX does not have the right entry for preprints. This is a hack for producing the correct reference:
@booklet{EasyChair:2820,
  author    = {Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego},
  title     = {Some Modular Considerations Regarding Odd Perfect Numbers - Part II},
  howpublished = {EasyChair Preprint 2820},
  year      = {EasyChair, 2020}}
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