Carmichael number

A composite natural number $n$ for which $a^{n-1} \equiv 1 \pmod{n}$ whenever $a$ is relatively prime to $n$. Thus they are pseudo-primes (cf. Pseudo-prime) for every such base $a$. These numbers play a role in the theory of probabilistic primality tests (cf. Probabilistic primality test), as they show that Fermat's theorem, to wit $ a^p \equiv a \pmod{p}$, whenever $p$ is prime and $a \not\equiv 0 \pmod{p}$, is not a sufficient criterion for primality (cf. also Fermat little theorem).

The first five Carmichael numbers are

\[561,\ 1105,\ 1729,\ 2465,\ 2821 \]

R.D. Carmichael [a2] characterized them as follows. Let $\lambda(n)$ be the exponent of the multiplicative group of integers modulo $n$, that is, the least positive $\lambda$ making all $\lambda$-th powers in the group equal to $1$. (This is readily computed from the prime factorization of $n$.) Then a composite natural number $n$ is Carmichael if and only if $\lambda(n) \mid n-1$. From this it follows that every Carmichael number is odd, square-free, and has at least $3$ distinct prime factors.

Let $C(x)$ denote the number of Carmichael numbers $\le x$. W.R. Alford, A. Granville and C. Pomerance [a1] proved that $C(x) > x^{2/7}$ for sufficiently large $x$. This settled a long-standing conjecture that there are infinitely many Carmichael numbers. It is believed on probabilistic grounds that $\log C(x) \sim \log x$. [a4].

P. Erdős proved in 1956 that $ C(X) < X \exp(- k \log X \log\log\log X / \log\log X) $ for some constant $ k $: he also gave a heuristic suggesting that his upper bound should be close to the true rate of growth of $ C(X) $.[a5]

There is apparently no better way to compute $C(x)$ than to make a list of the Carmichael numbers up to $x$. The most exhaustive computation to date (1996) is that of R.G.E. Pinch, who used the methods of [a3] to determine that $C\left({10^{16}}\right) = 246,683$.

References

[a1]	W.R. Alford, A. Granville, C. Pomerance, "There are infinitely many Carmichael numbers" Ann. of Math. , 140 (1994) pp. 703–722
[a2]	R.D. Carmichael, "Note on a new number theory function" Bull. Amer. Math. Soc. , 16 (1910) pp. 232–238 (See also: Amer. Math. Monthly 19 (1912), 22–27)
[a3]	R.G.E. Pinch, "The Carmichael numbers up to $10^{15}$" Math. Comp. , 61 (1993) pp. 381–391
[a4]	C. Pomerance, J.L. Selfridge, S.S. Wagstaff, Jr., "The pseudoprimes to $25.10^9$" Math. Comp. , 35 (1980) pp. 1003–1026
[a5]	P. Erdős, "On pseudoprimes and Carmichael numbers" Publ. Math. Debrecen', 4 (1956) pp.201–206.