# Carmichael number

*14 January 2011*.

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 |

## [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

- ↑ 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 - ↑ W. R. Alford, A. Granville, and C. Pomerance. "There are Infinitely Many Carmichael Numbers",
*Annals of Mathematics***139**(1994) 703-722. MR**95k**:11114 - ↑ Richard Guy, "Unsolved problems in Number Theory" (3rd ed), Springer-Verlag (2004) ISBN 0-387-20860-7. Section A13

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