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.

in celebration of his Sixtieth Birthday Let d be a square-free 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 3|h(−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 3|h(−d). Write N−(X) for the number of positive square-free d ≤ X for which 3|h(−d), and similarly let N+(X) be the number of positive square-free d ≤ X for which 3|h(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.

Carmichael numbers with three prime factors

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, non-primes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is

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.

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Building upon the work of Carl Pomerance and others, the central purpose of this discourse is to discuss the distribution of base-2 pseudoprimes, as well as improve upon Pomerance's conjecture regarding the Carmichael number counting function [8]. All conjectured formulas apply to any base b ≥ 2 for x ≥ x0(b). A table of base-2 pseudoprime, 2-strong pseudoprime, and Carmichael number counts up to 10 15 from [4] is included in the Appendix. We also discuss strong pseudoprimes and probabilistic primality testing. 1