Abstract
In this article, we revisit Carmichael numbers that are answers to the converse of Fermat’s little theorem. We take a slightly different constructive path from that of R D Carmichael in arriving at the first Carmichael number 561. We introduce Carmichael triplets and pose some interesting questions regarding the same.
This is a preview of subscription content, log in via an institution to check access.
Suggested Reading
L E Dickson, History of the Theory of Numbers, Chelsa Publishing Company, New York, 1952.
R D Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc., Vol.16, No.5, pp.232–238, 1910.
D M Burton, Elementary Number Theory, Sixth edn, McGraw-Hill, New York, 2007.
W Dunham, Journey through Genius, Wiley Science Editions, New York, 1990.
B Sury, The prime ordeal, Resonance, Vol.13, No.9, pp.866–881, 2008.
M V Yathirajsharma, Problem 12402, Am. Math. Mon., Vol.130, No.6, p.587, 2023.
W R Alford, Andrew Granville and Carl Pomerance, There are infinitely many Carmichael numbers, Ann. Math., Vol.139, No.2, pp.703–722, 1994.
R D Carmichael, On composite numbers P which satisfy the Fermat congruence aP−1 ≡ 1 (mod P), Am. Math. Mon., Vol.19, No.2, pp.22–27, 1912.
Acknowledgements
The author would like to thank the anonymous reviewer for the invaluable suggestions and rectification of mistakes, which really improved the quality of this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Yathirajsharma is currently working in Sarada Vilas College, Mysuru, Kamataka. He works primarily in the area of Number Theory. He likes observing the way one’s mind works while solving problems.
Rights and permissions
About this article
Cite this article
Yathirajsharma, M.V. A Revisit of Carmichael Numbers and a Note on Carmichael Triplets. Reson 29, 477–492 (2024). https://doi.org/10.1007/s12045-024-0477-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12045-024-0477-7