Some Modular Considerations Regarding Odd Perfect Numbers - Part II

EasyChair Preprint 2820

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 p^k 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)/p^k 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

@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}}