Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Odd perfect numbers are greater than $ 10^{1500}$


Authors: Pascal Ochem and Michaël Rao
Journal: Math. Comp. 81 (2012), 1869-1877
MSC (2010): Primary 11A25, 11A51
DOI: https://doi.org/10.1090/S0025-5718-2012-02563-4
Published electronically: January 30, 2012
MathSciNet review: 2904606
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Brent, Cohen, and te Riele proved in 1991 that an odd perfect number $ N$ is greater than $ 10^{300}$. We modify their method to obtain $ N>10^{1500}$. We also obtain that $ N$ has at least 101 not necessarily distinct prime factors and that its largest component (i.e. divisor $ p^a$ with $ p$ prime) is greater than $ 10^{62}$.


References [Enhancements On Off] (What's this?)

  • 1. R.P. Brent, G.L. Cohen, H.J.J. te Riele. Improved techniques for lower bounds for odd perfect numbers, Math. Comp. 57 (1991), no. 196, pp 857-868. MR 1094940 (92c:11004)
  • 2. G.L. Cohen. On the largest component of an odd perfect number, J. Austral. Math. Soc. Ser. A 42 (1987), pp 280-286. MR 869751 (87m:11005)
  • 3. T. Goto, Y. Ohno. Odd perfect numbers have a prime factor exceeding $ 10^8$, Math. Comp. 77 (2008), no. 263, pp 1859-1868. MR 2398799 (2009b:11008)
  • 4. K.G. Hare. New techniques for bounds on the total number of prime factors of an odd perfect number, Math. Comp. 76 (2007), no. 260, pp 2241-2248. MR 2336293 (2008g:11006)
  • 5. P.P. Nielsen. Odd perfect numbers have at least nine different prime factors, Math. Comp. 76 (2007), no. 160, pp 2109-2126. MR 2336286 (2008g:11153)
  • 6. T. Nagell. Introduction to Number Theory, John Wiley & Sons Inc., New York, 1951. MR 0043111 (13:207b)
  • 7. J.B. Rosser, L. Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp 64-94. MR 0137689 (25:1139)
  • 8. http://www.trnicely.net/pi/pix_0000.htm

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11A25, 11A51

Retrieve articles in all journals with MSC (2010): 11A25, 11A51


Additional Information

Pascal Ochem
Affiliation: LRI, CNRS, Bât 490 Université Paris-Sud 11, 91405 Orsay cedex, France
Email: ochem@lri.fr

Michaël Rao
Affiliation: CNRS, Lab J.V. Poncelet, Moscow, Russia. LaBRI, 351 cours de la Libération, 33405 Talence cedex, France
Email: rao@labri.fr

DOI: https://doi.org/10.1090/S0025-5718-2012-02563-4
Received by editor(s): March 27, 2011
Received by editor(s) in revised form: April 14, 2011
Published electronically: January 30, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society