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The Carmichael Numbers up to 10^15
, 1992
"... There are 105212 Carmichael numbers up to 10 : we describe the calculations. ..."
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There are 105212 Carmichael numbers up to 10 : we describe the calculations.
The RabinMonier theorem for Lucas pseudoprimes
 Math. Comp
, 1997
"... Abstract. We give bounds on the number of pairs (P, Q)with0≤P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P, Q). 1. ..."
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Cited by 3 (0 self)
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Abstract. We give bounds on the number of pairs (P, Q)with0≤P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P, Q). 1.
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.