# 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.

## Properties

- Every Carmichael number is square-free and has at least three different prime factors
- For every Carmichael number
*c*it holds that is divisible by for every one of its prime factors . - 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 , 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 .

## 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 Glyn Harman (2005),^{[6]} improving the earlier lower bound of obtained by Alford, Granville and Pomerance (1994), which first established that there were infinitely many Carmichael numbers.^{[7]}. The asymptotic rate of growth of *C*(*X*) is not known.^{[8]}

## References and notes

- ↑ J. Chernick, "On Fermat's simple theorem",
*Bull. Amer. Math. Soc.***45**(1939) 269-274 - ↑ (2003-11-22) Generic Carmichael Numbers
- ↑ Paulo Ribenboim,
*The new book of prime number records*, Springer-Verlag (1996) ISBN 0-387-94457-5. P.120 - ↑ Paul Erdős, "On pseudoprimes and Carmichael numbers",
*Publ. Math. Debrecen***4**(1956) 201-206. MR**18**18 - ↑ C. Pomerance, J.L. Selfridge and S.S. Wagstaff jr, "The pseudoprimes to 25.10
^{9}",*Math. Comp.***35**(1980) 1003-1026. MR**82g**:10030 - ↑ 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. - ↑ 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. - ↑ Richard Guy (2004).
*Unsolved problems in Number Theory*, 3rd. Springer-Verlag. ISBN 0-387-20860-7. . Section A13