# Carmichael number

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A Carmichael number is a composite number named after the mathematician Robert Daniel Carmichael. A Carmichael number $\scriptstyle c\$ divides $\scriptstyle a^c - a\$ for every integer $\scriptstyle a\$. A Carmichael number c also satisfies the congruence $\scriptstyle a^{c-1} \equiv 1 \pmod c$, if $\scriptstyle \operatorname{gcd}(a,c) = 1$. 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.

##  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 pn − 1 for every one of its prime factors pn.
• Every absolute Euler pseudoprime is a Carmichael number.

##  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 $\scriptstyle 6n+1\$, $\scriptstyle 12n+1\$ and $\scriptstyle 18n+1\$ are prime numbers, the product $\scriptstyle M_3(n) = (6n+1)\cdot (12n+1)\cdot (18n+1)$ is a Carmichael number. This condition can only be satisfied if the number $n\$ ends with 0, 1, 5 or 6. An equivalent formulation of Chernick's construction is that if $\scriptstyle m\$, $\scriptstyle 2m-1\$ and $\scriptstyle 3m-2$ are prime numbers, then the product $\scriptstyle m\cdot (2m-1)\cdot (3m-2)$ is a Carmichael number.

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

$M_k(n)=(6n+1)(12n+1)\prod_{i=1}^{k-2}(9\cdot 2^i n+1) \,$

with the condition that each of the factors is prime and that $n\$ is divisible by 2k − 4.

## 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

$X^{2/7} < C(X) < X \exp(-\log X \log\log\log X / \log\log X) . \,$

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]

## 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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