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Mathematics of Computation

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Carmichael numbers in the sequence $ (2^n k+1)_{n\geq 1}$

Authors: Javier Cilleruelo, Florian Luca and Amalia Pizarro-Madariaga
Journal: Math. Comp. 85 (2016), 357-377
MSC (2010): Primary 11A51, 11J86, 11J87
Published electronically: June 15, 2015
MathSciNet review: 3404453
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Abstract: We prove that for each odd number $ k$, the sequence $ (k2^n+1)_{n\ge 1}$ contains only a finite number of Carmichael numbers. We also prove that $ k=27$ is the smallest value for which such a sequence contains some Carmichael number.

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Additional Information

Javier Cilleruelo
Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, España

Florian Luca
Affiliation: School of Mathematics, University of the Witwatersrand, P. O. Box Wits 2050, South Africa

Amalia Pizarro-Madariaga
Affiliation: Instituto de Matemáticas, Universidad de Valparaiso, Chile

Keywords: Carmichael numbers, Subspace Theorem, linear forms in logarithms
Received by editor(s): August 12, 2013
Received by editor(s) in revised form: July 22, 2014, and July 29, 2014
Published electronically: June 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society