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Sieve methods for odd perfect numbers


Authors: S. Adam Fletcher, Pace P. Nielsen and Pascal Ochem
Journal: Math. Comp. 81 (2012), 1753-1776
MSC (2010): Primary 11A25; Secondary 11N36, 11A51, 11Y99
DOI: https://doi.org/10.1090/S0025-5718-2011-02576-7
Published electronically: January 9, 2012
MathSciNet review: 2904601
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Abstract | References | Similar Articles | Additional Information

Abstract: Using a new factor chain argument, we show that $ 5$ does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range $ 10^{8}<p<10^{1000}$. These results are generalized to much broader situations.


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Additional Information

S. Adam Fletcher
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: adam3.14159@gmail.com

Pace P. Nielsen
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: pace@math.byu.edu

Pascal Ochem
Affiliation: CNRS, Lab. J.V. Poncelet, Moscow LRI, Bât. 490, Univ. Paris-Sud 11, 91405, Orsay Cedex, France
Email: ochem@lri.fr

DOI: https://doi.org/10.1090/S0025-5718-2011-02576-7
Keywords: Abundance, factor chains, large sieve, odd perfect number
Received by editor(s): April 6, 2011
Received by editor(s) in revised form: May 27, 2011
Published electronically: January 9, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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