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 A014233 Smallest odd number for which Miller-Rabin primality test on bases <= n-th prime does not reveal compositeness. 4
 2047, 1373653, 25326001, 3215031751, 2152302898747, 3474749660383, 341550071728321, 341550071728321, 3825123056546413051, 3825123056546413051, 3825123056546413051, 318665857834031151167461, 3317044064679887385961981 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Note that some terms are repeated. Same as A006945 except for first term. a(12) > 2^64.  Hence the primality of numbers < 2^64 can be determined by asserting strong pseudoprimality to all prime bases <= 37 (=prime(12)). Testing to prime bases <=31 does not suffice, as a(11) < 2^64 and a(11) is a strong pseudoprime to all prime bases <= 31 (=prime(11)). - Joerg Arndt, Jul 04 2012 REFERENCES R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 157. LINKS Martin R. Albrecht, Jake Massimo, Kenneth G. Paterson, Juraj Somorovsky, Prime and Prejudice: Primality Testing Under Adversarial Conditions, Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, 281-298. Joerg Arndt, Matters Computational (The Fxtbook), section 39.10, pp. 786-792. Paul D. Beale, A new class of scalable parallel pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers, arXiv:1411.2484 [physics.comp-ph], 2014-2015. Paul D. Beale, Jetanat Datephanyawat, Class of scalable parallel and vectorizable pseudorandom number generators based on non-cryptographic RSA exponentiation ciphers, arXiv:1811.11629 [cs.CR], 2018. C. Caldwell, Strong probable-primality and a practical test. G. Jaeschke, On strong pseudoprimes to several bases, Mathematics of Computation, 61 (1993), 915-926. Yupeng Jiang, Yingpu Deng, Strong pseudoprimes to the first 9 prime bases, arXiv:1207.0063v1 [math.NT], June 30, 2012. A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see section 4.2.3, Miller-Rabin test. C. Pomerance, J. L. Selfridge and S. S. Wagstaff, Jr., The pseudoprimes to 25.10^9, Mathematics of Computation 35 (1980), pp. 1003-1026. Eric Bach, Explicit bounds for primality testing and related problems, Mathematics of Computation 55 (1990), pp. 355-380. F. Raynal, Miller-Rabin's Primality Test K. Reinhardt, Miller-Rabin Primality Test for odd n Jonathan P. Sorenson, Jonathan Webster, Strong Pseudoprimes to Twelve Prime Bases, arXiv:1509.00864 [math.NT], 2015. S. Wagon, Primality testing, Math. Intellig., 8 (No. 3, 1986), 58-61. Eric Weisstein's World of Mathematics, Strong Pseudoprime Eric Weisstein's World of Mathematics, Rabin-Miller Strong Pseudoprime Test Wikipedia, Miller-Rabin primality test Zhenxiang Zhang and Min Tang, Finding strong pseudoprimes to several bases. II, Mathematics of Computation 72 (2003), pp. 2085-2097. FORMULA Bach shows that, on the ERH, a(n) >> exp(sqrt(1/2 * x log x)). - Charles R Greathouse IV, May 17 2011 CROSSREFS Sequence in context: A258812 A321556 A321550 * A160964 A022193 A069386 Adjacent sequences:  A014230 A014231 A014232 * A014234 A014235 A014236 KEYWORD nonn,hard,more AUTHOR Jud McCranie, Feb 15 1997 EXTENSIONS Minor edits from N. J. A. Sloane, Jun 20 2009 a(9)-a(11) from Charles R Greathouse IV, Aug 14 2010 a(12)-a(13) from the Sorenson/Webster reference, Joerg Arndt, Sep 04 2015 STATUS approved

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Last modified December 5 18:14 EST 2020. Contains 338965 sequences. (Running on oeis4.)