Odd perfect numbers, Diophantine equations, and upper bounds
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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 $k\geq 10$.References
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Additional Information
- Pace P. Nielsen
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- MR Author ID: 709329
- Email: pace@math.byu.edu
- Received by editor(s): June 14, 2013
- Received by editor(s) in revised form: December 16, 2013
- Published electronically: February 18, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2549-2567
- MSC (2010): Primary 11N25; Secondary 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-2015-02941-X
- MathSciNet review: 3356038