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

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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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 𝑞𝑘𝑛2 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

  1. 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.
  2. Dickson, L. E. (1971) History of the Theory of Numbers, Vol. 1, Chelsea Pub. Co., New York, pp. 3–33.
  3. 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.
  4. Holdener, J. A. (2006) Conditions equivalent to the existence of odd perfect numbers, Math. Mag., 79, 389–391.
  5. Lustig, D. (2010) The algebraic independence of the sum of divisors functions, Journal of Number Theory, 130, 2628–2633.
  6. Sloane, N. J. A., OEIS sequence A033879 – Deficiency of 𝑛, or 2𝑛−𝜎(𝑛). Available online: http://oeis.org/A033879.
  7. 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

APA

Dris, 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.

Chicago

Dris, 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.

MLA

Dris, 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.

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