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Two contradictory conjectures concerning Carmichael numbers
"... Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to ..."
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Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in persontoperson discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].
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.
Theorem
"... in celebration of his Sixtieth Birthday Let d be a squarefree integer, which may be positive or negative, and let h(−d) be the class number of Q ( √ −d). In this paper we investigate the frequency of values of d for which 3h(−d). It follows from conjectures of Cohen and Lenstra [3], that asymptot ..."
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in celebration of his Sixtieth Birthday Let d be a squarefree integer, which may be positive or negative, and let h(−d) be the class number of Q ( √ −d). In this paper we investigate the frequency of values of d for which 3h(−d). It follows from conjectures of Cohen and Lenstra [3], that asymptotically a constant proportion of values of d have this property. The conjectured proportion is different for positive and negative d, being 1 − (1 − 3 −j) j=1 in the case of imaginary quadratics, for example. It follows from the work of Davenport and Heilbronn [5] that a positive proportion of d have 3 ∤ h(−d), both in the case of d positive and d negative. However it remains an open problem whether or not the same is true for values with 3h(−d). Write N−(X) for the number of positive squarefree d ≤ X for which 3h(−d), and similarly let N+(X) be the number of positive squarefree d ≤ X for which 3h(d). It was shown by Ankeny and Chowla [1] that N−(X) tends to infinity with X, and in fact their method yields N−(X) ≫ X 1/2. The best known result in this direction is that due to Soundararajan [7], who shows that N−(X) ≫ε X 7/8−ε, for any positive ε. In the case of real quadratic fields it was shown by Byeon and Koh [2] how Soundararajan’s analysis can be adapted to prove N+(X) ≫ε X 7/8−ε. The purpose of this note is to present a small improvement on these results, as follows.
Notes by G.J.O. Jameson
"... Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, nonprimes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is ..."
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Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, nonprimes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is
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.