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Mathematics of Computation
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
Published electronically: January 30, 2012
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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}$.


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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: http://dx.doi.org/10.1090/S0025-5718-2012-02563-4
PII: S 0025-5718(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.