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Sierpiński and Carmichael numbers


Authors: William Banks, Carrie Finch, Florian Luca, Carl Pomerance and Pantelimon Stănică
Journal: Trans. Amer. Math. Soc. 367 (2015), 355-376
MSC (2010): Primary 11J81, 11N25; Secondary 11A07, 11A51
Published electronically: September 23, 2014
MathSciNet review: 3271264
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Abstract: We establish several related results on Carmichael, Sierpiński and Riesel numbers. First, we prove that almost all odd natural numbers $ k$ have the property that $ 2^nk+1$ is not a Carmichael number for any $ n\in \mathbb{N}$; this implies the existence of a set $ \mathscr {K}$ of positive lower density such that for any $ k\in \mathscr {K}$ the number $ 2^nk+1$ is neither prime nor Carmichael for every $ n\in \mathbb{N}$. Next, using a recent result of Matomäki and Wright, we show that there are $ \gg x^{1/5}$ Carmichael numbers up to $ x$ that are also Sierpiński and Riesel. Finally, we show that if $ 2^nk+1$ is Lehmer, then $ n\le 150\,\omega (k)^2\log k$, where $ \omega (k)$ is the number of distinct primes dividing $ k$.


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Additional Information

William Banks
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: bbanks@math.missouri.edu

Carrie Finch
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Email: finchc@wlu.edu

Florian Luca
Affiliation: School of Mathematics, University of the Witwatersrand, P.O. Box Wits 2050, South Africa and Mathematical Institute, UNAM Juriquilla, Santiago de Querétaro 76230, Querétaro de Arteaga, México
Email: fluca@matmor.unam.mx

Carl Pomerance
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
Email: carl.pomerance@dartmouth.edu

Pantelimon Stănică
Affiliation: Department of Applied Mathematics, Naval Postgraduate School, Monterey, California 93943
Email: pstanica@nps.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06083-2
Received by editor(s): October 1, 2012
Received by editor(s) in revised form: January 16, 2013
Published electronically: September 23, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.