Estimating the counts of Carmichael and Williams numbers with small multiple seeds

Author:

Zhang Zhenxiang

Abstract

For a positive even integer L L , let P ( L ) \mathcal {P}(L) denote the set of primes p p for which p 1 p-1 divides L L but p p does not divide L L , let C ( L ) \mathcal {C}(L) denote the set of Carmichael numbers n n where n n is composed entirely of primes in P ( L ) \mathcal {P}(L) and where L L divides n 1 n-1 , and let W ( L ) C ( L ) \mathcal {W}(L)\subseteq \mathcal {C}(L) denote the subset of Williams numbers, which have the additional property that p + 1 n + 1 p+1 \mid n+1 for each prime p n p\mid n . We study | C ( L ) | |\mathcal {C}(L)| and | W ( L ) | |\mathcal {W}(L)| for certain integers L L . We describe procedures for generating integers L L that have more even divisors than any smaller positive integer, and we obtain certain numerical evidence to support the conjectures that log 2 | C ( L ) | = 2 s ( 1 + o ( 1 ) ) \log _2|\mathcal {C}(L)|=2^{s(1+o(1))} and log 2 | W ( L ) | = 2 s 1 / 2 o ( 1 ) \log _2|\mathcal {W}(L)|=2^{s^{1/2-o(1)}} when such an “even-divisor optimal” integer L L has s s different prime factors. For example, we determine that | C ( 735134400 ) | > 2 10 111 |\mathcal {C}(735134400)| > 2\cdot 10^{111} . Last, using a heuristic argument, we estimate that more than 2 24 2^{24} Williams numbers may be manufactured from a particular set of 1029 1029 primes, although we do not construct any explicit examples, and we describe the difficulties involved in doing so.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

Reference21 articles.

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2. There are infinitely many Carmichael numbers;Alford, W. R.;Ann. of Math. (2),1994

3. Korselt numbers and sets;Bouallègue, Kais;Int. J. Number Theory,2010

4. New primality criteria and factorizations of 2^{𝑚}±1;Brillhart, John;Math. Comp.,1975

5. Note on a new number theory function;Carmichael, R. D.;Bull. Amer. Math. Soc.,1910

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