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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.
Building Pseudoprimes With A Large Number Of Prime Factors
, 1995
"... 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 Dicks ..."
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Cited by 2 (0 self)
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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.
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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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.
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.
EXTENDED DICKSON POLYNOMIALS
, 1993
"... /=0 nA l J where [_J denotes the greatest integer function and x is an indeterminate, are commonly referred to as Dickson polynomials (e.g., see [6]). These polynomials have been studied in the past years, both from the point of view of their theoretical properties [2], [6], and [14], and from tha ..."
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/=0 nA l J where [_J denotes the greatest integer function and x is an indeterminate, are commonly referred to as Dickson polynomials (e.g., see [6]). These polynomials have been studied in the past years, both from the point of view of their theoretical properties [2], [6], and [14], and from that of their practical applications [7], [9], [10]. and [13]. In particular, their relevance to publickey cryptosystems has been pointed out in [8], [11], [12], and [16]. As is shown, e.g., in [14], the coefficients of pn(x, c) are integers for any positive integer n and c GZ. It is also evident that pn(x,\) = Vn(x \ (1.2) where Vn(x) = xVn_l{x) + Vn_2(x) [V0(x) = 2, Vx(x) = x] are the Lucas polynomials considered in [3] and [5]. In particular, we have P„Q,i) = L„, (1.3) where Ln is the /2 th Lucas number. In this paper, we consider the extended Dickson polynomials p„(x, c, U) defined in the next section. 2. INTRODUCTION AND DEFINITIONS