Keneth Adrian Precillas Dagal and Jose Arnaldo Bebita Dris
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 12—19
Download full paper: PDF, 203 Kb
Authors and affiliations
In this note, we show that if N = qkn2 is an odd perfect number with special prime q, and N is not divisible by 3, then the inequality q < n holds. We then give another unconditional proof for the inequality q < n which is independent of the results of Brown and Starni.
- Descartes–Frenicle–Sorli conjecture
- Odd perfect number
- Special prime
- Abundancy index
2020 Mathematics Subject Classification
- Acquaah, P., & Konyagin, S. (2012). On prime factors of odd perfect numbers, International Journal of Number Theory, 8, 1537–1540.
- Beasley, B. D. (2013). Euler and the ongoing search for odd perfect numbers. ACMS 19th Biennial Conference Proceedings, Bethel University, May 29 to Jun. 1, 2013.
- Brown, P. A. (2016). A partial proof of a conjecture of Dris. Preprint. Available online at: http://arxiv.org/abs/1602.01591.
- 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, http://arxiv.org/abs/1204.1450.
- Dris, J. A. B. (2012). The abundancy index of divisors of odd perfect numbers. Journal of Integer Sequences, 15, Article 12.4.4.
- Dris, J. A. B. (2017). On a curious biconditional involving the divisors of odd perfect numbers. Notes on Number Theory and Discrete Mathematics, 23(4), 1–13.
- Nielsen, P. P. (2007). Odd perfect numbers have at least nine distinct prime factors. Mathematics of Computation, 76, 2109–2126.
- Nielsen, P. P. (2015). Odd perfect numbers, Diophantine equations, and upper bounds. Mathematics of Computation, 84, 2549–2567.
- Ochem, P., & Rao, M. (2012). Odd perfect numbers are greater than 101500. Mathematics of Computation, 81, 1869–1877.
- Sorli, R. M. (2003). Algorithms in the Study of Multiperfect and Odd Perfect Numbers. Ph.D. Thesis, University of Technology, Sydney. Available online at: http://hdl.handle.net/10453/20034.
- Starni, P. (2018). On Dris conjecture about odd perfect numbers. Notes on Number Theory and Discrete Mathematics, 24(1), 5–9.
- Various contributors. Great Internet Mersenne Prime Search. Available online at: http://www.mersenne.org/, last accessed: 08/13/2020.
Cite this paper
Dagal, K. A. P., & Dris, J. A. B. (2021). The abundancy index of divisors of odd perfect numbers – Part II. Notes on Number Theory and Discrete Mathematics, 27(2), 12-19, doi: 10.7546/nntdm.2021.27.2.12-19.