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A014233 Smallest odd number for which Miller-Rabin primality test on bases <= n-th prime does not reveal compositeness. 3
2047, 1373653, 25326001, 3215031751, 2152302898747, 3474749660383, 341550071728321, 341550071728321, 3825123056546413051, 3825123056546413051, 3825123056546413051 (list; graph; refs; listen; history; text; internal format)



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]


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.


Table of n, a(n) for n=1..11.

Index entries for sequences related to pseudoprimes

Joerg Arndt, Matters Computational (The 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


Bach shows that, on the ERH, a(n) >> exp(sqrt(1/2 * x log x)). [Charles R Greathouse IV, May 17, 2011]


Sequence in context: A075950 A022527 A024009 * A160964 A022193 A069386

Adjacent sequences:  A014230 A014231 A014232 * A014234 A014235 A014236




Jud McCranie Feb 15 1997


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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Last modified October 30 10:33 EDT 2014. Contains 248797 sequences.