Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

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
Posted: January 9, 2012
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Abstract | References | Similar Articles | Additional Information

Abstract: Using a new factor chain argument, we show that 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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