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Mathematics of Computation

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Higher-order Carmichael numbers

Author: Everett W. Howe
Journal: Math. Comp. 69 (2000), 1711-1719
MSC (1991): Primary 11A51; Secondary 11N25, 11Y11, 13B40
Published electronically: February 17, 2000
MathSciNet review: 1709151
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We define a Carmichael number of order $m$ to be a composite integer $n$such that $n$th-power raising defines an endomorphism of every ${\mathbf Z}/n{\mathbf Z}$-algebra that can be generated as a ${\mathbf Z}/n{\mathbf Z}$-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 $2$.

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Additional Information

Everett W. Howe
Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967, USA

Keywords: Carmichael number, pseudoprime, \'etale algebra
Received by editor(s): December 7, 1998
Published electronically: February 17, 2000
Article copyright: © Copyright 2000 American Mathematical Society