Sierpiński and Carmichael numbers
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- by William Banks, Carrie Finch, Florian Luca, Carl Pomerance and Pantelimon Stănică PDF
- Trans. Amer. Math. Soc. 367 (2015), 355-376 Request permission
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 $n\leqslant 150 \omega (k)^2\log k$, where $\omega (k)$ is the number of distinct primes dividing $k$.References
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Additional Information
- William Banks
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 336964
- 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
- MR Author ID: 630217
- Email: fluca@matmor.unam.mx
- Carl Pomerance
- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
- MR Author ID: 140915
- Email: carl.pomerance@dartmouth.edu
- Pantelimon Stănică
- Affiliation: Department of Applied Mathematics, Naval Postgraduate School, Monterey, California 93943
- Email: pstanica@nps.edu
- Received by editor(s): October 1, 2012
- Received by editor(s) in revised form: January 16, 2013
- Published electronically: September 23, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 355-376
- MSC (2010): Primary 11J81, 11N25; Secondary 11A07, 11A51
- DOI: https://doi.org/10.1090/S0002-9947-2014-06083-2
- MathSciNet review: 3271264