A014233 Smallest odd number for which Miller-Rabin primality test on bases <= n-th prime does not reveal compositeness.

2047, 1373653, 25326001, 3215031751, 2152302898747, 3474749660383, 341550071728321, 341550071728321, 3825123056546413051, 3825123056546413051, 3825123056546413051, 318665857834031151167461, 3317044064679887385961981

Offset: 1

Author

Jud McCranie, Feb 15 1997

Keywords

• nonn
• hard
• more

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.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 98.

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.
• Index entries for sequences related to pseudoprimes

Formula

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

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