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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
##### Keywords:
Carmichael numbers
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##### References:
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