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 [broken link]
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
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