There are no Carmichael numbers of the form $2^np+1$ with $p$ prime

Adel Alahmadi; Florian Luca

doi:10.5802/crmath.393

Number theory

There are no Carmichael numbers of the form

2^{n} p + 1

with

p

prime

Adel Alahmadi ¹; Florian Luca ^2,^3,⁴

¹Research Group in Algebraic Structures and its Applications, King Abdulaziz University, Jeddah, Saudi Arabia
²Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico
³Research Group in Algebraic Structures and Applications, King Abdulaziz University, Abdulah Sulayman, Jeddah 22254, Saudi Arabia
⁴School of Maths, Wits University, 1 Jan Smuts, Braamfontein 2000, Johannesburg, South Africa

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1177-1181

Abstract

In this paper, we prove the theorem announced in the title.

Received: 2021-11-29
Revised: 2022-03-11
Accepted: 2022-06-08
Published online: 2022-10-04

DOI: 10.5802/crmath.393

Classification: 11A51

Author's affiliations:

Adel Alahmadi ¹ ; Florian Luca ^{2
,

3
,

4}

¹ Research Group in Algebraic Structures and its Applications, King Abdulaziz University, Jeddah, Saudi Arabia
² Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico
³ Research Group in Algebraic Structures and Applications, King Abdulaziz University, Abdulah Sulayman, Jeddah 22254, Saudi Arabia
⁴ School of Maths, Wits University, 1 Jan Smuts, Braamfontein 2000, Johannesburg, South Africa

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2022__360_G10_1177_0,
     author = {Adel Alahmadi and Florian Luca},
     title = {There are no {Carmichael} numbers of the form $2^np+1$ with $p$ prime},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1177--1181},
     year = {2022},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     doi = {10.5802/crmath.393},
     language = {en},
}

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Adel Alahmadi; Florian Luca. There are no Carmichael numbers of the form $2^np+1$ with $p$ prime. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1177-1181. doi: 10.5802/crmath.393

References
Cited by

[1] William R. Alford; Andrew Granville; Carl Pomerance There are infinitely many Carmichael numbers, Ann. Math., Volume 139 (1994) no. 3, pp. 703-722 | MR | Zbl | DOI

[2] Williams D. Banks; Carrie Finch; Florian Luca; Carl Pomerance; Pantelimon Stănică Sierpiński and Carmichael numbers, Trans. Am. Math. Soc., Volume 367 (2015) no. 1, pp. 355-376 | DOI | Zbl

[3] Javier Cilleruelo; Florian Luca; Amalia Pizarro Carmichael numbers in the sequence ${(2^{n} k + 1)}_{n \geq 1}$ , Math. Comput., Volume 85 (2016) no. 297, pp. 357-377 | MR | Zbl | DOI

[4] Paul Erdős; Andrew M. Odlyzko On the density of odd integers of the form $(p - 1) 2^{- n}$ and related questions, J. Number Theory, Volume 11 (1979), pp. 257-263 | MR | DOI | Zbl

[5] Andrew Granville Primes in intervals of bounded length, Bull. Am. Math. Soc., Volume 52 (2015) no. 2, pp. 171-222 | MR | DOI | Zbl

[6] Thomas Wright The impossibility of certain types of Carmichael numbers, Integers, Volume 12 (2012) no. 5, pp. 951-964 | MR | Zbl | DOI

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