Higher-order Carmichael numbers, by Everett W. Howe
This paper has now appeared in Math. Comp. 69 (2000), 1711-1719,
and so the preprint has been removed.
We define a Carmichael number of order m to be a composite
integer n such that nth-power raising defines an
endomorphism of every Z/nZ-algebra that can be
generated as a Z/nZ-module by m elements.
We give a simple criterion to determine whether a number is a Carmichael
number of order m, and we give a heuristic argument (based on an
argument of Erdos for the usual Carmichael numbers) that indicates that
for every m there should be infinitely many Carmichael numbers of
order m. The argument suggests a method for finding examples of
higher-order Carmichael numbers; we use the method to provide examples
of Carmichael numbers of order 2.
Everett W. Howe <email@example.com>