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HigherOrder Carmichael Numbers
 Math. Comp
, 1998
"... . We define a Carmichael number of order m to be a composite integer n such that nthpower raising defines an endomorphism of every Z=nZalgebra that can be generated as a Z=nZmodule by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and w ..."
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. We define a Carmichael number of order m to be a composite integer n such that nthpower raising defines an endomorphism of every Z=nZalgebra that can be generated as a Z=nZmodule 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 higherorder Carmichael numbers; we use the method to provide examples of Carmichael numbers of order 2. 1. Introduction A Carmichael number is defined to be a positive composite integer n that is a Fermat pseudoprime to every base; that is, a composite n is a Carmichael number if a n j a mod n for every integer a. Clearly one can generalize the idea of a Carmichael number by allowing the pseudoprimality test in the definition to vary over some...
MATHEMATICS OF COMPUTATION
, 2000
"... Abstract. We define a Carmichael number of order m to be a composite integer n such that nthpower raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZmodule by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, an ..."
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Abstract. We define a Carmichael number of order m to be a composite integer n such that nthpower raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZmodule 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 Erdős 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 higherorder Carmichael numbers; we use the method to provide examples of Carmichael numbers of order 2. 1.
ABSOLUTE QUADRATIC PSEUDOPRIMES
"... Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1. ..."
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Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1.