# Carmichael number  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] Code [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

A Carmichael number is a composite number named after the mathematician Robert Daniel Carmichael. A Carmichael number $c\$ divides $a^{c}-a\$ for every integer $a\$ . A Carmichael number c also satisfies the congruence $a^{c-1}\equiv 1{\pmod {c}}$ , if $\operatorname {gcd} (a,c)=1$ . 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 $p_{n}-1$ for every one of its prime factors $p_{n}$ .
• Every absolute Euler pseudoprime is a Carmichael number.

## Chernick's Carmichael numbers

J. Chernick found in 1939 a way to construct Carmichael numbers . If, for a natural number n, the three numbers $6n+1\$ , $12n+1\$ and $18n+1\$ are prime numbers, the product $M_{3}(n)=(6n+1)\cdot (12n+1)\cdot (18n+1)$ 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 $m\$ , $2m-1\$ and $3m-2$ are prime numbers, then the product $m\cdot (2m-1)\cdot (3m-2)$ is a Carmichael number.

This way to construct Carmichael numbers may be extended 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 $2^{k-4}$ .

## 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^{0.332} The upper bound is due to Erdős(1956) and Pomerance, Selfridge and Wagstaff (1980) and the lower bound is due to Glyn Harman (2005), improving the earlier lower bound of $X^{2/7}$ obtained by Alford, Granville and Pomerance (1994), which first established that there were infinitely many Carmichael numbers.. The asymptotic rate of growth of C(X) is not known.

## 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. Glyn Harman (2005). "On the number of Carmichael numbers up to x". Bulletin of the London Mathematical Society 37: 641–650. DOI:10.1112/S0024609305004686. Zbl. 1108.11065. Research Blogging.
7. W. R. Alford, A. Granville, and C. Pomerance (1994). "There are Infinitely Many Carmichael Numbers". Annals of Mathematics 139: 703-722. MR 95k:11114 Zbl 0816.11005.
8. Richard Guy (2004). Unsolved problems in Number Theory, 3rd. Springer-Verlag. ISBN 0-387-20860-7. . Section A13