Generalizations
The notion of Carmichael number generalizes to a Carmichael ideal in any number field K. For any nonzero prime ideal in, we have for all in, where is the norm of the ideal . (This generalizes Fermat's little theorem, that for all integers m when p is prime.) Call a nonzero ideal in Carmichael if it is not a prime ideal and for all, where is the norm of the ideal . When K is, the ideal is principal, and if we let a be its positive generator then the ideal is Carmichael exactly when a is a Carmichael number in the usual sense.
When K is larger than the rationals it is easy to write down Carmichael ideals in : for any prime number p that splits completely in K, the principal ideal is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in . For example, if p is any prime number that is 1 mod 4, the ideal (p) in the Gaussian integers Z is a Carmichael ideal.
Both prime and Carmichael numbers satisfy the following equality:
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