Sieve Methods for Odd Perfect Numbers

S. Adam Fletcher 1 P. Nielsen Pace 1 Pascal Ochem 2
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
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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Contributor : Pascal Ochem <>
Submitted on : Saturday, October 6, 2012 - 6:32:53 PM
Last modification on : Thursday, May 24, 2018 - 3:59:22 PM

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S. Adam Fletcher, P. Nielsen Pace, Pascal Ochem. Sieve Methods for Odd Perfect Numbers. Mathematics of Computation, American Mathematical Society, 2012, 81, pp.1753-1776. ⟨lirmm-00739250⟩

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