Comptes Rendus
Number theory
There are no Carmichael numbers of the form 2 n p+1 with p prime
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1177-1181.

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

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.393
Classification: 11A51
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},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.393/

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

[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) n1 , Math. Comput., Volume 85 (2016) no. 297, pp. 357-377 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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