On Dris conjecture about odd perfect numbers

Paolo Starni
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 1, Pages 5—9
DOI: 10.7546/nntdm.2018.24.1.5-9
Download full paper: PDF, 141 Kb

Details

Authors and affiliations

Paolo Starni
School of Economics, Management, and Statistics
Rimini Campus, University of Bologna
Via Anghera 22, 47921 Rimini, Italy

Abstract

The Euler’s form of odd perfect numbers, if any, is n = παN2, where π is prime, (π, N) = 1 and πα ≡ 1 (mod 4). Dris conjecture states that N > πα. We find that N2 > 1/2 πγ, with γ = max{ω(n) − 1, α}; ω(n) ≥ 10 is the number of distinct prime factors of n.

Keywords

  • Odd perfect numbers
  • Dris conjecture

2010 Mathematics Subject Classification

  • 11A05
  • 11A25

References

  1. Brown, P. (2016) A partial proof of a conjecture of Dris, http://arxiv.org/abs/ 1602.01591v1.
  2. Chen, S. C., & Luo, H. (2011) Odd multiperfect numbers, http://arxiv.org/abs/ 1102.4396.
  3. Dickson, L. E. (2005) History of the Theory of Numbers, Vol. 1, Dover, New York.
  4. Dris, J. A. B. (2008), Solving the odd perfect number problem: some old and new approaches, M.Sc. thesis, De La Salle University, Manila, http://arxiv.org/abs/ 1204.1450.
  5. MacDaniel, W. L., & Hagis, P. (1975) Some results concerning the non-existence of odd perfect numbers of the form παM2β, Fibonacci Quart., 131, 25–28.
  6. Nielsen, P. P. (2015) Odd perfect numbers, Diophantine equations, and upper bounds, Math. Comp., 84, 2549–2567.
  7. Sorli, R. M. (2003) Algorithms in the study of multiperfect and odd perfect numbers, Ph.D. thesis, University of Technology, Sydney, http://epress.lib.uts.edu.au. /research/handle/10453/20034.
  8. Starni, P. (1991) On the Euler’s factor of an odd perfect number, J. Number Theory, 37, 366–369.
  9. Starni, P. (1993) Odd perfect numbers: a divisor related to the Euler’s factor, J. Number Theory, 44, 58–59.
  10. Starni, P. (2006) On some properties of the Euler’s factor of certain odd perfect numbers, J. Number Theory, 116, 483–486.

Related papers

Cite this paper

APA

Starni, P. (2018). On Dris conjecture about odd perfect numbers. Notes on Number Theory and Discrete Mathematics, 24(1), 5-9, doi: 10.7546/nntdm.2018.24.1.5-9.

Chicago

Starni, Paolo. “On Dris Conjecture about Odd Perfect Numbers.” Notes on Number Theory and Discrete Mathematics 24, no. 1 (2018): 5-9, doi: 10.7546/nntdm.2018.24.1.5-9.

MLA

Starni, Paolo. “On Dris Conjecture about Odd Perfect Numbers.” Notes on Number Theory and Discrete Mathematics 24.1 (2018): 5-9. Print, doi: 10.7546/nntdm.2018.24.1.5-9.

Comments are closed.