Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 3, Pages 62—67

DOI: 10.7546/nntdm.2018.24.3.62-67

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## Details

### Authors and affiliations

Jose Arnaldo Bebita Dris

*Institute of Mathematics, University of the Philippines
Carlos P. Garcia Avenue, Diliman, Quezon City, Philippines
*

Doli-Jane Uvales Tejada

*Mathematics Department, College of Natural Sciences and Mathematics
Mindanao State University, General Santos City, Philippines
*

### Abstract

The Descartes–Frenicle–Sorli conjecture predicts that 𝑘 = 1 if 𝑞^{𝑘}𝑛^{2} is an odd perfect number with Euler prime 𝑞. In this note, we present some further conditions equivalent to this conjecture..

### Keywords

- Odd perfect number
- Abundancy index
- Deficiency

### 2010 Mathematics Subject Classification

- 11A25

### References

- Beasley, B. D. (2013) Euler and the ongoing search for odd perfect numbers, Proc. of ACMS 19-th Biennial Conference Proceedings, Bethel University, 29 May–1 June, 2013, pp. 21–31. Available online: http://godandmath.files.wordpress.com/2013/07/

acms-2013-proceedings.pdf. - Dickson, L. E. (1971) History of the Theory of Numbers, Vol. 1, Chelsea Pub. Co., New York, pp. 3–33.
- Dris, J. A. B. (2017) Conditions equivalent to the Descartes–Frenicle–Sorli conjecture on odd perfect numbers, Notes on Number Theory and Discrete Mathematics, 23 (2), 12–20.
- Holdener, J. A. (2006) Conditions equivalent to the existence of odd perfect numbers, Math. Mag., 79, 389–391.
- Lustig, D. (2010) The algebraic independence of the sum of divisors functions, Journal of Number Theory, 130, 2628–2633.
- Sloane, N. J. A., OEIS sequence A033879 – Deficiency of 𝑛, or 2𝑛−𝜎(𝑛). Available online: http://oeis.org/A033879.
- Sorli, R. M. (2003) Algorithms in the study of multiperfect and odd perfect numbers, Ph. D. Thesis, University of Technology, Sydney.

## Related papers

## Cite this paper

APADris, J. A. B., & Tejada, D.-J. U. (2018). Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers – Part II. Notes on Number Theory and Discrete Mathematics, 24(3), 62-67, doi: 10.7546/nntdm.2018.24.3.62-67.

ChicagoDris, Jose Arnaldo Bebita, and Doli-Jane Uvales Tejada. “Conditions Equivalent to the Descartes–Frenicle–Sorli Conjecture on Odd Perfect Numbers – Part II.” Notes on Number Theory and Discrete Mathematics 24, no. 3 (2018): 62-67, doi: 10.7546/nntdm.2018.24.3.62-67.

MLADris, Jose Arnaldo Bebita, and Doli-Jane Uvales Tejada. “Conditions Equivalent to the Descartes–Frenicle–Sorli Conjecture on Odd Perfect Numbers – Part II.” Notes on Number Theory and Discrete Mathematics 24.3 (2018): 62-67. Print, doi: 10.7546/nntdm.2018.24.3.62-67.