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Carmichael Numbers of the form (6m + 1)(12m + 1)(18m + 1)
, 2002
"... Numbers of the form (6m + 1)(12m + 1)(18m + 1) where all three factors are simultaneously prime are the best known examples of Carmichael numbers. In this paper we tabulate the counts of such numbers up to 10 for each n 42. We also derive a function for estimating these counts that is remarkably ..."
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Numbers of the form (6m + 1)(12m + 1)(18m + 1) where all three factors are simultaneously prime are the best known examples of Carmichael numbers. In this paper we tabulate the counts of such numbers up to 10 for each n 42. We also derive a function for estimating these counts that is remarkably accurate.
On using Carmichael numbers for public key encryption systems
, 1997
"... We show that the inadvertent use of a Carmichael number instead of a prime factor in the modulus of an RSA cryptosystem is likely to make the system fatally vulnerable, but that such numbers may be detected. ..."
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We show that the inadvertent use of a Carmichael number instead of a prime factor in the modulus of an RSA cryptosystem is likely to make the system fatally vulnerable, but that such numbers may be detected.
On Generalized Carmichael Numbers
, 2000
"... . For arbitrary integers k 2 Z we investigate the set C k of the generalized Carmichael numbers, i.e. the natural numbers n ? maxf1; 1 \Gamma kg such that the equation a n+k j a mod n holds for all a 2 N. We give a characterization of these generalized Carmichael numbers and discuss several spe ..."
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. For arbitrary integers k 2 Z we investigate the set C k of the generalized Carmichael numbers, i.e. the natural numbers n ? maxf1; 1 \Gamma kg such that the equation a n+k j a mod n holds for all a 2 N. We give a characterization of these generalized Carmichael numbers and discuss several special cases. In particular, we prove that C 1 is finite and that C k is infinite, whenever 1 \Gamma k ? 1 is squarefree. We also discuss generalized Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers n which satisfy the equation a n j a mod n only for a = 2, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares. 1 Introduction: Historical Background On October 18th, 1640, Pierre de Fermat wrote in a letter to Bernard Frenicle de Bessy that if p is a prime number, then p divides a p\Gamma1 \Gamma 1 for all in...
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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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.
NEW POLYNOMIALS PRODUCING ABSOLUTE PSEUDOPRIMES WITH ANY NUMBER OF PRIME FACTORS
, 2007
"... Abstract. In this paper, we introduce a certain method to construct polynomials producing many absolute pseudoprimes. By this method, we give new polynomials producing absolute pseudoprimes with any fixed number of prime factors which can be viewed as a generalization of Chernick’s result. By the si ..."
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Abstract. In this paper, we introduce a certain method to construct polynomials producing many absolute pseudoprimes. By this method, we give new polynomials producing absolute pseudoprimes with any fixed number of prime factors which can be viewed as a generalization of Chernick’s result. By the similar method, we give another type of polynomials producing many absolute pseudoprimes. As concrete examples, we tabulate the counts of such numbers of our forms. 1.
unknown title
, 711
"... Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implement ..."
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Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implementation up to 10 24. Statistics relevant to the distribution of threeprime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [10]. 1.
unknown title
, 711
"... Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implement ..."
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Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implementation up to 10 24. Statistics relevant to the distribution of threeprime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [10]. 1.
GENERATING MSTRONG FIBONACCI PSEUDOPRIMES Adina Di Porto and
, 1991
"... One of the most important problems to be faced when using publickey cryptosystems (see [7] for background material) is to generate a large number of large (> 10 1 0 0) prime numbers. This hard to handle problem has been elegantly bypassed by submitting randomly generated odd integers n (which are, ..."
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One of the most important problems to be faced when using publickey cryptosystems (see [7] for background material) is to generate a large number of large (> 10 1 0 0) prime numbers. This hard to handle problem has been elegantly bypassed by submitting randomly generated odd integers n (which are, of course,