Abstract not found

There are 105212 Carmichael numbers up to 10 : we describe the calculations.

We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.

. 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. 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...

Abstract. 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/nZalgebra 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 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 higher-order Carmichael numbers; we use the method to provide examples of Carmichael numbers of order 2. 1.

Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1.