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Estimating the counts of Carmichael and Williams numbers with small multiple seeds. (English) Zbl 1315.11101
Summary: For a positive even integer $$L$$, let $$\mathcal {P}(L)$$ denote the set of primes $$p$$ for which $$p-1$$ divides $$L$$ but $$p$$ does not divide $$L$$, let $$\mathcal {C}(L)$$ denote the set of Carmichael numbers $$n$$ where $$n$$ is composed entirely of primes in $$\mathcal {P}(L)$$ and where $$L$$ divides $$n-1$$, and let $$\mathcal {W}(L)\subseteq \mathcal {C}(L)$$ denote the subset of Williams numbers, which have the additional property that $$p+1 \mid n+1$$ for each prime $$p\mid n$$. We study $$|\mathcal {C}(L)|$$ and $$|\mathcal {W}(L)|$$ for certain integers $$L$$. We describe procedures for generating integers $$L$$ that have more even divisors than any smaller positive integer, and we obtain certain numerical evidence to support the conjectures that $$\log _2|\mathcal {C}(L)|=2^{s(1+o(1))}$$ and $$\log _2|\mathcal {W}(L)|=2^{s^{1/2-o(1)}}$$ when such an “even-divisor optimal” integer $$L$$ has $$s$$ different prime factors. For example, we determine that $$|\mathcal {C}(735134400)| > 2\cdot 10^{111}$$. Last, using a heuristic argument, we estimate that more than $$2^{24}$$ Williams numbers may be manufactured from a particular set of $$1029$$ primes, although we do not construct any explicit examples, and we describe the difficulties involved in doing so.
##### MSC:
 11Y16 Number-theoretic algorithms; complexity 11A51 Factorization; primality 11Y11 Primality
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##### References:
 [1] Alaoglu, L.; Erd{\"o}s, P., On highly composite and similar numbers, Trans. Amer. Math. Soc., 56, 448-469 (1944) · Zbl 0061.07903 [2] Alford, W. R.; Granville, Andrew; Pomerance, Carl, There are infinitely many Carmichael numbers, Ann. of Math. (2), 139, 3, 703-722 (1994) · Zbl 0816.11005 [3] Bouall{\e}gue, Kais; Echi, Othman; Pinch, Richard G. E., Korselt numbers and sets, Int. J. Number Theory, 6, 2, 257-269 (2010) · Zbl 1214.11013 [4] Brillhart, John; Lehmer, D. H.; Selfridge, J. L., New primality criteria and factorizations of $$2^m\pm 1$$, Math. Comp., 29, 620-647 (1975) · Zbl 0311.10009 [5] Carmichael, R. D., Note on a new number theory function, Bull. Amer. Math. Soc., 16, 5, 232-238 (1910) · JFM 41.0226.04 [6] Chen, Zhuo; Greene, John, Some comments on Baillie-PSW pseudoprimes, Fibonacci Quart., 41, 4, 334-344 (2003) · Zbl 1052.11006 [7] Crandall, Richard; Pomerance, Carl, Prime numbers, xvi+597 pp. (2005), Springer: New York:Springer · Zbl 1088.11001 [8] Dusart, Pierre, The $$k$$ th prime is greater than $$k(\ln k+\ln \ln k-1)$$ for $$k\geq 2$$, Math. Comp., 68, 225, 411-415 (1999) · Zbl 0913.11039 [9] Echi, Othman, Williams numbers, C. R. Math. Acad. Sci. Soc. R. Can., 29, 2, 41-47 (2007) · Zbl 1204.11185 [10] Erd{\"o}s, P., On highly composite numbers, J. London Math. Soc., 19, 130-133 (1944) · Zbl 0061.07904 [11] Erd{\"o}s, P., On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen, 4, 201-206 (1956) · Zbl 0074.27105 [12] [Gran1992] A. Granville, Primality testing and Carmichael numbers, Notices of the American Mathematical Society 39 (1992), 696-700. [13] [Kor] A. Korselt, Probl\“eme chinois, L”interm\'ediaire des math\'ematiciens 6 (1899), 142-143. [14] Nicolas, Jean-Louis, R\'epartition des nombres hautement compos\'es de Ramanujan, Canad. J. Math., 23, 116-130 (1971) · Zbl 0213.06602 [15] [Pom84] C. Pomerance, Are there counter-examples to the Baillie-PSW primality test?, Dopo Le Parole aangeboden aan Dr. A. K. Lenstra (H. W. Lenstra, Jr., J. K. Lenstraand P. Van Emde Boas, eds.), Amsterdam, 1984. [16] Ramanujan, S., Highly composite numbers [Proc. London Math. Soc. (2) {\bf 14} (1915), 347-409]. Collected papers of Srinivasa Ramanujan, 78-128 (2000), AMS Chelsea Publ., Providence, RI [17] Robin, G., M\'ethodes d’optimisation pour un probl\`“eme de th\'”eorie des nombres, RAIRO Inform. Th\'eor., 17, 3, 239-247 (1983) · Zbl 0531.10012 [18] Rosen, Kenneth H., Elementary number theory and its applications, xviii+638 pp. (2000), Addison-Wesley: Reading, MA:Addison-Wesley · Zbl 0964.11002 [19] Williams, H. C., On numbers analogous to the Carmichael numbers, Canad. Math. Bull., 20, 1, 133-143 (1977) · Zbl 0368.10011 [20] Zhang, Zhenxiang, A one-parameter quadratic-base version of the Baillie-PSW probable prime test, Math. Comp., 71, 240, 1699-1734 (2002) · Zbl 1076.11063 [21] Zhang, Zhenxiang, Counting Carmichael numbers with small seeds, Math. Comp., 80, 273, 437-442 (2011) · Zbl 1225.11161
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