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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Carrie Finch
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450

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

Carl Pomerance
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551

Pantelimon Stănică
Affiliation: Department of Applied Mathematics, Naval Postgraduate School, Monterey, California 93943

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.

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