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Higher-order Carmichael numbers. (English) Zbl 0966.11006
By generalizing a characterization of a Carmichael number $$n$$ by means of cyclic $$\mathbb{Z}/n\mathbb{Z}$$-modules the author defined a Carmichael number of order $$m$$ to be a composite integer $$n>0$$ such that $$n$$th-power raising defines an endomorphism of every $$\mathbb{Z}/n \mathbb{Z}$$-algebra that can be generated as a $$\mathbb{Z}/n \mathbb{Z}$$-module by $$m$$ elements. These numbers can be described by the conditions a) $$n$$ is squarefree, b) for every prime divisor $$p$$ of $$n$$ and for every integer $$r$$ with $$1\leq r\leq m$$, there is an integer $$i\geq 0$$ such that $$n\equiv p^i\bmod (p^r-1)$$. The main part of the paper is devoted to the introduction of a method which at least in the case $$m=2$$ allows to construct some Carmichael numbers of order $$m$$. Finally a characterization of a special type of such numbers by means of finite étale algebras is given.

##### MSC:
 11A51 Factorization; primality 11N25 Distribution of integers with specified multiplicative constraints 11Y11 Primality 13B40 Étale and flat extensions; Henselization; Artin approximation
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