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Counting Carmichael numbers with small seeds. (English) Zbl 1225.11161
Let \(A_s\) be the product of the first \(s\) primes (called seeds by the author), \({\mathcal P}_s\) the set of primes \(p\) for which \(p-1\) divides \(A_s\) but \(p\) does not divide \(A_s\), \({\mathcal C}_s\) the set of Carmichael numbers \(n\) such that \(n\) is composed entirely of the primes in \({\mathcal P}_s\) and such that \(A_s\) divides \(n-1\). The author gives numerical evidence for the conjecture: \[ |{\mathcal C}_s|=2^{2^{s(1-\varepsilon)}}, \lim_{s\to\infty}\varepsilon=0. \] This shows that \(|{\mathcal C}_s|\) grows rapidly with \(s\). He describes a procedure to compute exact values of \(|{\mathcal C}_s|\) for small \(s\).

MSC:
11Y11 Primality
11Y16 Number-theoretic algorithms; complexity
11Y35 Analytic computations
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