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A014233
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Smallest odd number for which Miller-Rabin primality test on bases <= n-th prime does not reveal compositeness.
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3
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2047, 1373653, 25326001, 3215031751, 2152302898747, 3474749660383, 341550071728321, 341550071728321, 3825123056546413051, 3825123056546413051, 3825123056546413051
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OFFSET
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1,1
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COMMENTS
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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]
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REFERENCES
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R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 157.
S. Wagon, Primality testing, The Mathematical Intelligencer 8:3 (1986), pp. 58-61.
Zhenxiang Zhang and Min Tang, "Finding strong pseudoprimes to several bases. II", Mathematics of Computation 72 (2003), pp. 2085-2097.
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LINKS
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Table of n, a(n) for n=1..11.
Index entries for sequences related to pseudoprimes
Joerg Arndt, Fxtbook, section 39.10, pp. 786-792
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
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Eric Weisstein's World of Mathematics, Rabin-Miller Strong Pseudoprime Test
Wikipedia, Miller-Rabin primality test
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FORMULA
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Bach shows that, on the ERH, a(n) >> exp(sqrt(1/2 * x log x)). [Charles R Greathouse IV, May 17, 2011]
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CROSSREFS
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Sequence in context: A075950 A022527 A024009 * A160964 A022193 A069386
Adjacent sequences: A014230 A014231 A014232 * A014234 A014235 A014236
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KEYWORD
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nonn,hard,more,changed
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AUTHOR
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Jud McCranie Feb 15 1997
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EXTENSIONS
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Minor edits from N. J. A. Sloane, Jun 20 2009
a(9)-a(11) from Charles R Greathouse IV, Aug 14 2010
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STATUS
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approved
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