An alternative and equivalent definition of Carmichael numbers is given by Korselt's theorem from 1899, which says that a positive integer N is a Carmichael number if and only if N is squarefree and nonprime, and for all prime divisors p of N, p1 divides N1, written as p1  N1. This shows that Carmichael numbers are always odd.
Carmichael numbers are sometimes called also absolute pseudoprimes.
The first known Carmichael number is 561. (Indeed, 561 = 3 · 11 · 17 is squarefree and 2  560, 10  560 and 16  560.) The next Carmichael numbers are (SIDN A002997) (http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002997):
Korselt was the first who observed the properties of these numbers but he could not get an example. In 1910 Carmichael[?] gave the example of 561 and hence the name. Paul Erdös heuristically argued there should be infinitely many Carmichael numbers. In 1994 it was shown by William Alford, Andrew Granville and Carl Pomerance that there really exist infinitely many Carmichael numbers. Richard G. E. Pinch also gave and proved an upper bound for C(n), the number of Carmichael numbers less than n.

Carmichael numbers have at least three positive prime factors. The first Carmichael numbers with k= 3, 4, 5, ... prime factors are (SIDN A006931) (http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=006931):
k  

3  561 = 3 · 11 · 17 
4  41041 = 7 · 11 · 13 · 41 
5  825265 = 5 · 7 · 17 · 19 · 73 
6  321197185 = 5 · 19 · 23 · 29 · 37 · 137 
7  5394826801 = 7 · 13 · 17 · 23 · 31 · 67 · 73 
8  232250619601 = 7 · 11 · 13 · 17 · 31 · 37 · 73 · 163 
9  9746347772161 = 7 · 11 · 13 · 17 · 19 · 31 · 37 · 41 · 641 
The first Carmichael numbers with 4 prime factors are:
n  

1  41041 = 7 · 11 · 13 · 41 
2  62745 = 3 · 5 · 47 · 89 
3  63973 = 7 · 13 · 19 · 37 
4  75361 = 11 · 13 · 17 · 31 
5  101101 = 7 · 11 · 13 · 101 
6  126217 = 7 · 13 · 19 · 73 
7  172081 = 7 · 13 · 31 · 61 
8  188461 = 7 · 13 · 19 · 109 
9  278545 = 5 · 17 · 29 · 113 
10  340561 = 13 · 17 · 23 · 67 
Higherorder Carmichael numbers
Carmichael numbers can be generalized using concepts of abstract algebra.
The above definition states that a composite integer N is Carmichael precisely when the Nthpowerraising function p_{N} from the ring Z_{N} of integers modulo N to itself is the identity function. The identity is the only Z_{N}algebra endomorphism on Z_{N} so we can restate the definition as asking that p_{N} be an algebra endomorphism of Z_{N}. As above, p_{N} satisfies the same property whenever N is prime.
The Nthpowerraising function p_{N} is also defined on any Z_{N}algebra A. A theorem states that N is prime if and only if all such functions p_{N} are algebra endomorphisms.
Inbetween these two conditions lies the definition of Carmichael number of order m for any positive integer m as any composite number N such that p_{N} is an endomorphism on every Z_{N}algebra that can be generated as Z_{N}module by m elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.
Korselt's criterion can be generalized to higherorder Carmichael numbers, see Howe's paper listed below.
A heuristic argument, given in the same paper, appears to suggest that there are infinitely many Carmichael numbers of order m, for any m. However, not a single Carmichael number of order 3 or above is known.
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