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

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## 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

- Broughan, K. A., Delbourgo, D., & Zhou, Q. (2013). Improving the Chen and Chen result for odd perfect numbers, Integers, 13, Article #A39.
- Brown, P. A. (2016). A partial proof of a conjecture of Dris, preprint, https://arxiv.org/abs/1602.01591.
- Chen, S.-C., & Luo, H. (2013). Odd multiperfect numbers, Bull. Aust. Math. Soc., 88 (1), 56–63.
- Cohen, G. L., & Sorli, R. M. (2012). On odd perfect numbers and even 3-perfect numbers, Integers, 12A, Article #A6.
- Dandapat, G. G., Hunsucker, J. L., & Pomerance, C. (1975). Some new results on odd perfect numbers, Pacific J. Math., 57 (2), 359–364.
- Dris, J. A. B. (2008). Solving the odd perfect number problem: some old and new approaches, M. Sc. thesis, De La Salle University, Manila, Philippines.
- 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.
- Dris, J. A. B., & MathStackExchange user FredH (2019). If is an odd perfect number with special prime , then is not a square. https://math.stackexchange.com/questions/3121498.
- Ewell, J. A. Jr. (1980). On the multiplicative structure of odd perfect numbers, Journal of Number Theory, 12, 339–342.
- Ochem, P. (2019). Answer to a question of the first author in Mathematics StackExchange, https://math.stackexchange.com/a/3151412/28816.
- Sloane, N. J. A., OEIS sequence A033879 – Deficiency of
*n*, or 2*n*−*σ*(*n*), https://oeis.org/A033879. - Sloane, N. J. A., & Guy, R. K., OEIS sequence A001065 – Sum of proper divisors (or aliquot parts) of
*n*: sum of divisors of*n*that are less than*n*, https://oeis.org/A001065. - Starni, P. (2018). On Dris conjecture about odd perfect numbers, Notes Number Theory Discrete Math., 24 (1), 5–9.
- Starni, P. (1991). On the Euler’s factor of an odd perfect number, Journal of Number Theory, 37, 366–369.
- Wikipedia contributors. (2019, March 6). Perfect number. In Wikipedia, The Free Encyclopedia. Retrieved from https://en.wikipedia.org/w/index.php?title=Perfect_number&oldid=886493275.

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## 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.