Revisiting some old results on odd perfect numbers

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 4, Pages 18–25
DOI: 10.7546/nntdm.2018.24.4.18-25
Full paper (PDF, 177 Kb)

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

In this note, we revisit and show how some old results on odd perfect numbers follow from assuming some unproven yet reasonable conjectures.

Keywords

• Odd perfect number
• Descartes–Frenicle–Sorli conjecture
• Dris conjecture
• Abundancy index
• Deficiency

2010 Mathematics Subject Classification

• 11A25

References

1. Beasley, B. D. (2013) Euler and the ongoing search for odd perfect numbers, ACMS 19th Biennial Conference Proceedings, Bethel University.
2. Brown, P. A. (2016) A partial proof of a conjecture of Dris, preprint, https://arxiv.org/pdf/1602.01591v1.pdf.
3. Dickson, L. E. (1971) History of the theory of numbers, Vol. 1, 3–33, Chelsea Pub. Co., New York.
4. Dris, J. A. B. (2008) Solving the odd perfect number problem: Some old and new approaches, M. S. Math thesis, De La Salle University, Manila, Philippines.
5. Dris, J. A. B. (2009) Solving the odd perfect number problem: Some new approaches, Electr. Proc. of the 11th Science and Technology Congress, ed. L. Pajo, De La Salle University.
6. Dris, J. A. B. (2012) The abundancy index of divisors of odd perfect numbers, J. Integ. Seq., 15 (4), Article 12.4.4.
7. 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.
8. Dris, J. A. B. (2017) On a curious biconditional involving divisors of odd perfect numbers. Notes on Number Theory and Discrete Mathematics, 23 (4), 1–13.
9. Dris, J. A. B., & Luca, F. (2016) A note on odd perfect numbers, Fibonacci Quart., 54 (4), 291–295.
10. Mihailescu, P. (2004) Primary cyclotomic units and a proof of Catalan’s Conjecture, J. Reine Angew. Math., 572, 167–195.
11. Nielsen, P. (2015) Odd perfect numbers, Diophantine equations, and upper bounds, Math. Comp., 84, 2549–2567.
12. Ochem, P., & Rao, M. (2012) Odd perfect numbers are greater than 10¹⁵⁰⁰, Math. Comp., 81, 1869–1877.
13. Sorli, R. M. (2003) Algorithms in the study of multiperfect and odd perfect numbers, Ph. D. Thesis, University of Technology, Sydney.
14. Starni, P. (2018) On Dris conjecture about odd perfect numbers. Notes on Number Theory and Discrete Mathematics, 23 (1), 5–9.