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Odd perfect numbers, Diophantine equations, and upper bounds


Author: Pace P. Nielsen
Journal: Math. Comp. 84 (2015), 2549-2567
MSC (2010): Primary 11N25; Secondary 11Y50
DOI: https://doi.org/10.1090/S0025-5718-2015-02941-X
Published electronically: February 18, 2015
MathSciNet review: 3356038
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Abstract: We obtain a new upper bound for odd multiperfect numbers. If $ N$ is an odd perfect number with $ k$ distinct prime divisors and $ P$ is its largest prime divisor, we find as a corollary that $ 10^{12}P^{2}N<2^{4^{k}}$. Using this new bound, and extensive computations, we derive the inequality $ k\geq 10$.


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Additional Information

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

DOI: https://doi.org/10.1090/S0025-5718-2015-02941-X
Keywords: Diophantine equation, perfect number
Received by editor(s): June 14, 2013
Received by editor(s) in revised form: December 16, 2013
Published electronically: February 18, 2015
Article copyright: © Copyright 2015 American Mathematical Society