Recent developments in primality testing
 Carl Pomerance
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Is there an algorithm that can decide whethern is prime or composite and that runs in polynomial time? The AdlemanRumely algorithm and the Lenstra variations come so close, that it would seem that almost any improvement would give the final breakthrough.
Is factoring inherently more difficult than distinguishing between primes and composites? Most people feel that this is so, but perhaps this problem too will soon yield.
In his “Disquisitiones Arithmeticae” Gauss [9], p. 396, wrote
“The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and prolix that even for numbers that do not exceed the limits of tables constructed by estimable men, i.e., for numbers that do not yield to artificial methods, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers.”
The struggle continues!
 L. M. Adleman, C. Pomerance, R. S. Rumely, On distinguishing prime numbers from composite numbers (to appear). Also see the paper by L. M. Adleman of the same title in Proc. 21st FOCS Conf. (1980)
 N. C. Ankeny, The least quadratic non residue. Ann. Math. (2) 55 (1952), 65–72. CrossRef
 J. Brillhart, D. H. Lehmer, J. L. Selfridge, New primality criteria and factorizations of 2^{m} ± 1. Math. Comp. 29 (1975), 620–647.
 R. A. DeMillo, R. J. Lipton, A. J. Perlis, Social processes and proofs of theorems and programs, Comm. ACM 22 (1979), 271–280. Also in Math. Intelligencer 3 (1980), 3140 CrossRef
 L. E. Dickson,History of the Theory of Numbers, Vol. 1, Press of Gibson Brothers, Washington, D.C. 1919
 J. D. Dixon, Asymptotically fast factorization of integers. Math. Comp. 36 (1981), 255–260 CrossRef
 P. Erdös, On almost primes, Amer. Math. Monthly 57 (1950), 404–407 CrossRef
 P. Erdös, S. Ulam, Some probabilistic remarks on Fermat’s last theorem. Rocky Mountain J. Math. 1 (1971), 613–616 CrossRef
 C. F. Gauss,Disquisitiones Arithmeticae (A. A. Clarke, transi.). Yale University Press, New Haven, Connecticut, London 1966
 D. B. Gillies, Three new Mersenne primes and a statistical theory. Math. Comp. 18 (1964), 93–97 CrossRef
 G. B. Kolata, Mathematical proofs: The genesis of reasonable doubt. Science 192 (1976), 989–990 CrossRef
 G. B. Kolata, Prior restraints on cryptogryphy considered. Science 208 (1980), 1442–1443. Also, Cryptology: A secret meeting at IDA? Science 200 (1978), 184. Also, D. Shapley and G. B. Kolata, Cryptology: Scientists puzzle over threat to open research, publication. Science 197 (1977), 13451349 CrossRef
 I. Lakatos,Proofs and Refutations. Cambridge University Press, Cambridge, London, New York, Melbourne 1976 CrossRef
 D. H. Lehmer, On the exact number of primes less than a given limit. Illinois J. Math. 3 (1959), 381–388
 D. H. Lehmer, Strong Carmichael numbers. J. Austral. Math. Soe. Ser. A 21 (1976), 508–510 CrossRef
 H. W. Lenstra, Jr., Miller’s primality test. Inform. Process. Lett. 8 (1979), 86–88 CrossRef
 H. W. Lenstra, Jr., Primality testing. Studieweek Getaltheorie en Computers, Sept. 15, 1980, Stichting Math. Centrum, Amsterdam
 E. Lucas,Théorie des Nombres. Tome 1, Librarie Blanchard, Paris, 1961
 D. E. G. Malm, On MonteCarlo primality tests. Unpublished (but see [25], Theorem 4)
 G. L. Miller, Riemann’s hypothesis and tests for primality. J. Comput. System Sci. 13 (1976), 300–317 CrossRef
 M. A. Morrison, J. Brillhart, A method of factoring and the factorization of F_{7}. Math. Comp. 29 (1975), 183–205
 C. Noll, L. Nickel, The 25th and 26th Mersenne primes. Math. Comp. 35 (1980), 1387–1390 CrossRef
 C. Pomerance, A new lower bound for the pseudoprime counting function. Illinois J. Math., to appear
 C. Pomerance, On the distribution of pseudoprimes. Math. Comp., to appear
 C. Pomerance, J. L. Selfridge, S. S. Wagstaff, Jr., The pseudoprimes to 25 · 10^{9}. Math. Comp. 35 (1980), 1003–1026
 V. R. Pratt, Every prime has a succinct certificate. SIAM J. Comput. 4 (1975), 214–220 CrossRef
 M. O. Rabin, Probabilistic algorithms. In:J. F. Traub, ed.,Algorithms and Complexity. Academic Press, New York, San Francisco, London, 1976, 21–39
 M. O. Rabin, Probabilistic algorithm for primality testing. J. Number Theory 12 (1980), 128–138 CrossRef
 R. Rivest, A. Shamir, L. Adleman, A method for obtaining digital signatures and publickey cryptosystems. MIT Lab for Comp. Sci., Technical Memo LCS/TM82,1977. Also Comm. ACM 21 (1978), 120–128 CrossRef
 J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94
 G. J. Simmons, Cryptology: The mathematics of secure communication. Math. Intelligencer 1 (1979), 233–246 CrossRef
 D. Slowinski, Searching for the 27th Mersenne prime. J. Recreational Math. 11 (197879), 258–261
 R. Solovay, V. Strassen, A fast MonteCarlo test for primality. SIAM J. Comput. 6 (1977), 84–85; erratum, 7 (1978), 118 CrossRef
 B. Tuckerman, The 24th Mersenne prime. Proc. Nat. Acad. Sci. USA 68 (1971), 2319–2320 CrossRef
 H. C. Williams, Primality testing on a computer. Ars Combinatoria 5 (1978), 127–185
 H. C. Williams, R. Holte, Some observations on primality testing, Math. Comp. 32 (1978), 905–917 CrossRef
 H. C. Williams, J. S. Judd, Some algorithms for prime testing using generalized Lehmer functions. Math. Comp. 30 (1976), 867–886
 M. Wunderlich, A report on the factorization of 2797 numbers using the continued fraction method. To appear
 G. J. Chaitin, J. T. Schwartz, A note on Monte Carlo primality tests and algorithmic information theory, Comm. Pure Appl. Math. 31 (1978), 521–527 CrossRef
 Title
 Recent developments in primality testing
 Journal

The Mathematical Intelligencer
Volume 3, Issue 3 , pp 97105
 Cover Date
 19810901
 DOI
 10.1007/BF03022861
 Print ISSN
 03436993
 Publisher
 SpringerVerlag
 Additional Links
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 Industry Sectors
 Authors

 Carl Pomerance ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Georgia, 30602, Athens, Georgia