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An asymptotic formula for the number of smooth values of a polynomial
 J. Number Theory
, 1999
"... Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from mult ..."
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Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity
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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Cited by 1 (0 self)
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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.