Summary
We discuss some simple deterministic algorithms that establish primality for a robust set of primes in polynomial time. The first 6 sections comprise the intact original article published in the first edition of this volume in 1997. The last 2 sections discuss developments in this fast-moving field to early 2013, and refer to the prior sections in the past tense. The bibliography for the original article and the new update have been combined.
For Paul Erdős on his eightieth birthday
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Supported in part by the Cultural Initiative Fund and the Russian Academy of Natural Sciences
- 2.
Supported in part by an NSF grant
References
L. M. Adleman and M.-D. A. Huang, Primality testing and two-dimensional abelian varieties over finite fields, Lecture Notes in Math. 1512, Springer-Verlag, Berlin, 1992, 142 pp.
L. M. Adleman, C. Pomerance, and R. S. Rumely, On distinguishing prime numbers from composite numbers, Ann. of Math. 117 (1983), 173–206.
M. Agrawal, N. Kayal, and N. Saxena, PRIMES is in P, Ann. of Math. 160 (2004), 781–793.
W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. 140 (1994), 703–722.
D. Bernstein, Proving primality in essentially quartic random time, Math. Comp. 76 (2007), 389–403.
W. Bosma and M.-P. van der Hulst, Primality proving with cyclotomy, Ph.D. thesis, Amsterdam (1990).
J. Brillhart, M. Filaseta, and A. Odlyzko, On an irreducibility theorem of A. Cohn, Can. J. Math. 33 (1981), 1055–1099.
J. Brillhart, D. H. Lehmer and J. L. Selfridge, New primality criteria and factorizations of 2m ± 1, Math. Comp. 29 (1975), 620–647.
J. P. Buhler, R. E. Crandall and M. A. Penk, Primes of the form n! ± 1 and 2 ⋅3 ⋅5…p ± 1, Math. Comp. 38 (1982), 639–643.
E. R. Canfield, P. Erdős and C. Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), 1–28.
R. Crandall and C. Pomerance, Prime numbers: a computational perspective, 2nd ed., Springer, New York, 2005.
H. Davenport, Multiplicative number theory, 2nd ed., Springer-Verlag, New York, 1980.
P. Erdős, On almost primes, Amer. Math. Monthly 57 (1950), 404–407.
P. Erdős, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), 201–206.
P. Erdős and J. H. van Lint, On the number of positive integers ≤ x and free of prime factors > y, Simon Stevin 40 (1966/67), 73–76.
M. R. Fellows and N. Koblitz, Self-witnessing polynomial-time complexity and prime factorization, Designs, Codes and Cryptography 2 (1992), 231–235.
M. Fürer, Deterministic and Las Vegas primality testing algorithms, in Proceedings of ICALP 85 (July 1985). Nafplion, Greece. W. Brauer, ed., Lecture Notes in Computer Science 194, Springer-Verlag, Berlin, 1985, pp. 199–209.
A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.
H. W. Lenstra, Jr., Miller’s primality test, Information Processing Letters 8 (1979), 86–88.
H. W. Lenstra, jr and Carl Pomerance, Primality testing with Gaussian periods, www.math.dartmouth.edu/~carlp/aks041411.pdf
F. Pappalardi, On Artin’s conjecture for primitive roots, Ph.D. thesis, McGill University (1993).
J. Pintz, W. L. Steiger and E. Szemerédi, Infinite sets of primes with fast primality tests and quick generation of large primes, Math. Comp. 53 (1989), 399–406.
C. Pomerance, Very short primality proofs, Math. Comp. 48 (1987), 315–322.
C. Pomerance, Primality testing: variations on a theme of Lucas, Congressus Numerantium 201 (2010), 301–312.
V. R. Pratt, Every prime has a succinct certificate, SIAM J. Comput. 4 (1975), 214–220.
J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.
R. Solovay and V. Strassen, A fast Monte Carlo test for primality, SIAM J. Comput. 6 (1977) 84–85; erratum 7 (1978), 118.
I. M. Vinogradov, On the bound of the least non-residue of nth powers, Bull. Acad. Sci. USSR 20 (1926), 47–58 (= Trans. Amer. Math. Soc. 29 (1927), 218–226).
B. Źrałek, A deterministic version of Pollard’s p − 1 algorithm, Math. Comp. 79 (2010), 513–533.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Konyagin, S., Pomerance, C. (2013). On Primes Recognizable in Deterministic Polynomial Time. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_12
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7258-2_12
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7257-5
Online ISBN: 978-1-4614-7258-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)