**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 3, Pages 8–24

DOI: 10.7546/nntdm.2020.26.3.8-24

**Full paper (PDF, 192 Kb)**

**Download Corrigendum: PDF, 159 Kb)**

## Details

### 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

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*(*m*^{2}) (modulo 8) are attained, if *p ^{k}m*

^{2}is an odd perfect number with special prime

*p*. We prove that

*σ*(*m*^{2}) ≡ 1 (mod 8) holds only if*p*+*k*≡ 2 (mod 16).*σ*(*m*^{2}) ≡ 3 (mod 8) holds only if*p*−*k*≡ 4 (mod 16).*σ*(*m*^{2}) ≡ 5 (mod 8) holds only if*p*+*k*≡ 10 (mod 16).*σ*(*m*^{2}) ≡ 7 (mod 8) holds only if*p*−*k*≡ 4 (mod 16).

We express gcd(*m*^{2}; *σ*(*m*^{2})) as a linear combination of *m*^{2} and *σ*(*m*^{2}). We also consider some applications under the assumption that *σ*(*m*^{2}) / *p ^{k}* is a square. Lastly, we prove a last-minute conjecture under this hypothesis.

### 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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## Corrigendum

- Dris, J. A. B., & San Diego, I. T. (2023). Corrigendum to: “Some modular considerations regarding odd perfect numbers – Part II” [Notes on Number Theory and Discrete Mathematics, 2020, Vol. 26, No. 3, 8–24].
*Notes on Number Theory and Discrete Mathematics*, 29(1), 181–184.

## Related papers

- Dris, J. A. B. (2017). On a curious biconditional involving the divisors of odd perfect numbers,
*Notes Number Theory Discrete Math*., 23(4), 1–13. - Dris, J. A. B. (2017). Conditions equivalent to the Descartes–Frenicle–Sorli conjecture on odd perfect numbers,
*Notes Number Theory Discrete Math*., 23(2), 12–20. - Dris, J. A.B., & San Diego, I.T. (2020). Some modular considerations regarding odd perfect numbers,
*Notes Number Theory Discrete Math*., 26(2), 27–33. - Starni, P. (2018). On Dris conjecture about odd perfect numbers,
*Notes Number Theory Discrete Math.*, 24(1), 5–9.

## Cite this paper

Dris, J. A. B., & San Diego, I. T. (2020). Some modular considerations regarding odd perfect numbers – Part II. *Notes on Number Theory and Discrete Mathematics*, 26(3), 8-24, DOI: 10.7546/nntdm.2020.26.3.8-24.