Odd perfect numbers are greater than $10^{1500}$
HTML articles powered by AMS MathViewer
- by Pascal Ochem and Michaël Rao PDF
- Math. Comp. 81 (2012), 1869-1877 Request permission
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
- R. P. Brent, G. L. Cohen, and H. J. J. te Riele, Improved techniques for lower bounds for odd perfect numbers, Math. Comp. 57 (1991), no. 196, 857–868. MR 1094940, DOI 10.1090/S0025-5718-1991-1094940-3
- Graeme L. Cohen, On the largest component of an odd perfect number, J. Austral. Math. Soc. Ser. A 42 (1987), no. 2, 280–286. MR 869751
- Takeshi Goto and Yasuo Ohno, Odd perfect numbers have a prime factor exceeding $10^8$, Math. Comp. 77 (2008), no. 263, 1859–1868. MR 2398799, DOI 10.1090/S0025-5718-08-02050-4
- Kevin G. Hare, New techniques for bounds on the total number of prime factors of an odd perfect number, Math. Comp. 76 (2007), no. 260, 2241–2248. MR 2336293, DOI 10.1090/S0025-5718-07-02033-9
- Pace P. Nielsen, Odd perfect numbers have at least nine distinct prime factors, Math. Comp. 76 (2007), no. 260, 2109–2126. MR 2336286, DOI 10.1090/S0025-5718-07-01990-4
- Trygve Nagell, Introduction to Number Theory, John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951. MR 0043111
- J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
- http://www.trnicely.net/pi/pix_0000.htm
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
- MR Author ID: 714149
- Email: rao@labri.fr
- Received by editor(s): March 27, 2011
- Received by editor(s) in revised form: April 14, 2011
- Published electronically: January 30, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 1869-1877
- MSC (2010): Primary 11A25, 11A51
- DOI: https://doi.org/10.1090/S0025-5718-2012-02563-4
- MathSciNet review: 2904606