Carmichael number

A Carmichael number is a composite number named after the mathematician Robert Daniel Carmichael. A Carmichael number divides for every integer . A Carmichael number c also satisfies the congruence , if . The first few Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601 and 8911. In 1994 Pomerance, Alford and Granville proved that there exist infinitely many Carmichael numbers.

Contents 1 Properties 2 Chernick's Carmichael numbers 3 Distribution of Carmichael numbers 4 References and notes

[edit] Properties

• Every Carmichael number is square-free and has at least three different prime factors
• For every Carmichael number c it holds that c − 1 is divisible by p_n − 1 for every one of its prime factors p_n.
• Every absolute Euler pseudoprime is a Carmichael number.

[edit] Chernick's Carmichael numbers

J. Chernick found in 1939 a way to construct Carmichael numbers^{[1] [2]}. If, for a natural number n, the three numbers , and are prime numbers, the product is a Carmichael number. This condition can only be satisfied if the number ends with 0, 1, 5 or 6. An equivalent formulation of Chernick's construction is that if , and are prime numbers, then the product is a Carmichael number.

This way to construct Carmichael numbers may be extended^[3] to

with the condition that each of the factors is prime and that is divisible by 2^{k − 4}.

[edit] Distribution of Carmichael numbers

Let C(X) denote the number of Carmichael numbers less than or equal to X. Then for all sufficiently large X

The upper bound is due to Erdős(1956)^[4] and Pomerance, Selfridge and Wagstaff (1980)^[5] and the lower bound is due to Alford, Granville and Pomerance (1994)^[6]. The asymptotic rate of growth of C(X) is not known.^[7]

[edit] References and notes

1. ↑ J. Chernick, "On Fermat's simple theorem", Bull. Amer. Math. Soc. 45 (1939) 269-274
2. ↑ (2003-11-22) Generic Carmichael Numbers
3. ↑ Paulo Ribenboim, The new book of prime number records, Springer-Verlag (1996) ISBN 0-387-94457-5. P.120
4. ↑ Paul Erdős, "On pseudoprimes and Carmichael numbers", Publ. Math. Debrecen 4 (1956) 201-206. MR 18 18
5. ↑ C. Pomerance, J.L. Selfridge and S.S. Wagstaff jr, "The pseudoprimes to 25.10⁹", Math. Comp. 35 (1980) 1003-1026. MR 82g:10030
6. ↑ W. R. Alford, A. Granville, and C. Pomerance. "There are Infinitely Many Carmichael Numbers", Annals of Mathematics 139 (1994) 703-722. MR 95k:11114
7. ↑ Richard Guy, "Unsolved problems in Number Theory" (3rd ed), Springer-Verlag (2004) ISBN 0-387-20860-7. Section A13

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