Conditions equivalent to the Descartes–Frenicle–Sorli Conjecture on odd perfect numbers

Jose Arnaldo B. Dris
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 2, Pages 12—20
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Authors and affiliations

Jose Arnaldo B. Dris
Department of Mathematics and Physics,
Far Eastern University
Nicanor Reyes Street, Sampaloc, Manila, Philippines


The Descartes-Frenicle-Sorli conjecture predicts that k = 1 if qkn2 is an odd perfect number with Euler prime q. In this note, we present some conditions equivalent to this conjecture.


  • Odd perfect number
  • Abundancy index
  • Deficiency

AMS Classification

  • 11A25


  1. 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.
  2. Broughan, K. A., Delbourgo, D. & Zhou, Q. (2013) Improving the Chen and Chen result for odd perfect numbers, Integers, 13, #A39.
  3. Brown, P. A. (2016) A partial proof of a conjecture of Dris, preprint ,
  4. Chen, F-J., & Chen, Y-G. (2012) On odd perfect numbers, Bull. Aust. Math. Soc., 86, 510–514.
  5. Chen, F-J. & Chen, Y-G. (2014) On the index of an odd perfect number, Colloq. Math., 136, 41–49.
  6. Dris, J. A. B. (2008) Solving the Odd Perfect Number Problem: Some Old and New Approaches, M. Sc. Math thesis, De La Salle University, Manila, Philippines.
  7. Dris, J. A. B. (2012) The abundancy index of divisors of odd perfect numbers, J. Integer Seq., 15, Article 12.4.4.
  8. Dris, J. A. B. & F. Luca (2016) A note on odd perfect numbers, Fibonacci Quart., 54(4), 291–295.
  9. Woltman, G. & Kurowski, S. The Great Internet Mersenne Prime Search, Last viewed: September 9, 2016.
  10. Holdener, J. A. (2006) Conditions equivalent to the existence of odd perfect numbers, Math. Mag., 79, 389–391.
  11. Sloane, N. J. A. OEIS sequence A033879 – Deficiency of n, or 2n − σ(n),
  12. Adajar, C. F. E. OEIS sequence A271816 – Deficient-perfect numbers: Deficient numbers n such that n/(2n − σ(n)) is an integer,
  13. Slowak, J. (1999) Odd perfect numbers, Math. Slovaca, 49 , 253–254.
  14. Sorli, R. M. (2003) Algorithms in the Study of Multiperfect and Odd Perfect Numbers, Ph. D. Thesis, University of Technology, Sydney,

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Cite this paper

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.

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