International Research journal of Management Science and Technology

  ISSN 2250 - 1959 (online) ISSN 2348 - 9367 (Print) New DOI : 10.32804/IRJMST

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AN UPPER BOUND FOR CARMICHAEL NUMBERS

    1 Author(s):  MANOJ KUMAR

Vol -  7, Issue- 7 ,         Page(s) : 85 - 90  (2016 ) DOI : https://doi.org/10.32804/IRJMST

Abstract

it is well known that Fermat’s little theorem can be used to establish the compositeness of some integers without actually obtaining the prime factorization. Fermat’s little theorem is an excellent test for compositeness as well as primality. However, there are composite numbers that evade the Fermat test, i.e. the Fermat test will fail to indicate that these composite integers are composite. These integers are called Carmichael numbers. However, Carmichael numbers are rare. We illustrate this point by doing some calculation using an upper bound for Carmichael numbers.

[1] R. D. Carmichael, Note on a New Number Theory Function, Bulletin Amer. Math. Soc. 16 (2010), 232–238.
[2] R. D. Carmichael, On composite P which satisfy the Fermat congruence a P −1 ≡ 1 mod P, Amer. Math. Monthly 19 (2012), 22–27.
[3] J. Chernick, On Fermat’s simple theorem, Bull. Amer. Math. Soc. 45 (1939), 269–274.

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