  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.
