# Carmichael number

The composite integer *n* is a **Carmichael number** if
*a*^{n-1}=1 (mod *n*) for every integer
*a* relatively prime to *n*. (This condition is
satisfied by all primes because of Fermat's Little Theorem.)
The Fermat probable primality test will fail to show a
Carmichael number is composite until we run across one of
its factors!

The Carmichael numbers under 100,000 are

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, and 75361.

Small Carmichael numbers are rare: there are only 2,163 are less than 25,000,000,000. (Recently, Richard Pinch has found that there are still only 246,683 Carmichael numbers below 10,000,000,000,000,000.) Nevertheless, in 1994 it was proved that there are infinitely many of them!

**See Also:** Pseudoprime, PRP

**Related pages** (outside of this work)

**References:**

- AGP94 (Infinitely many Carmichael numbers)
W. R. Alford,A. GranvilleandC. Pomerance, "There are infinitely many Carmichael numbers,"Ann. of Math. (2),139(1994) 703--722.MR 95k:11114- GP2001
A. GranvilleandC. Pomerance, "Two contradictory conjectures concerning Carmichael numbers,"Math. Comp.,71(2002) 883--908.MR 1 885 636(Abstract available)- Pinch93
R. Pinch, "The Carmichael numbers up to 10^{15},"Math. Comp.,61:203 (1993) 381-391.MR 93m:11137[A preprint and several data files may be found in the Carmichael directory of his FTP site. For example, he lists the Carmichaels to 10^{17}.]

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