Density of Carmichael numbers with three prime factors

Balasubramanian, R.; Nagaraj, S.

doi:10.1090/S0025-5718-97-00857-0

Density of Carmichael numbers with three prime factors
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by R. Balasubramanian and S. V. Nagaraj PDF

Math. Comp. 66 (1997), 1705-1708 Request permission

Abstract:

We get an upper bound of $O(x^{5/14+o(1)})$ on the number of Carmichael numbers $\leq x$ with exactly three prime factors.

References

W. R. Alford, Andrew Granville, and Carl Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703–722. MR 1283874, DOI 10.2307/2118576
W. R. Alford, Andrew Granville, and Carl Pomerance, On the difficulty of finding reliable witnesses, Algorithmic number theory (Ithaca, NY, 1994) Lecture Notes in Comput. Sci., vol. 877, Springer, Berlin, 1994, pp. 1–16. MR 1322705, DOI 10.1007/3-540-58691-1_{3}6
Ivan Damgård, Peter Landrock, and Carl Pomerance, Average case error estimates for the strong probable prime test, Math. Comp. 61 (1993), no. 203, 177–194. MR 1189518, DOI 10.1090/S0025-5718-1993-1189518-9
A. Granville, Primality testing and Carmichael numbers, Notices Amer. Math. Soc. 39 (1992), 696–700.
Neal Koblitz, A course in number theory and cryptography, Graduate Texts in Mathematics, vol. 114, Springer-Verlag, New York, 1987. MR 910297, DOI 10.1007/978-1-4684-0310-7
Mohan Nair, Multiplicative functions of polynomial values in short intervals, Acta Arith. 62 (1992), no. 3, 257–269. MR 1197420, DOI 10.4064/aa-62-3-257-269
R. G. E. Pinch, The Carmichael numbers up to $10^{15}$, Math. Comp. 61 (1993), no. 203, 381–391. MR 1202611, DOI 10.1090/S0025-5718-1993-1202611-7
Carl Pomerance, Carmichael numbers, Nieuw Arch. Wisk. (4) 11 (1993), no. 3, 199–209. MR 1251482
Carl Pomerance, J. L. Selfridge, and Samuel S. Wagstaff Jr., The pseudoprimes to $25\cdot 10^{9}$, Math. Comp. 35 (1980), no. 151, 1003–1026. MR 572872, DOI 10.1090/S0025-5718-1980-0572872-7
S. Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. (2) 14 (1915), 347–409.
Paulo Ribenboim, The book of prime number records, 2nd ed., Springer-Verlag, New York, 1989. MR 1016815, DOI 10.1007/978-1-4684-0507-1