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Some observations on primality testing


Authors: H. C. Williams and R. Holte
Journal: Math. Comp. 32 (1978), 905-917
MSC: Primary 10A25; Secondary 10-04
DOI: https://doi.org/10.1090/S0025-5718-1978-0476625-0
MathSciNet review: 0476625
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Abstract: Let N be an integer which is to be tested for primality. Previous methods of ascertaining the primality of N make use of factors of $ N \pm 1$, $ {N^2} \pm N + 1$, and $ {N^2} + 1$ in order to increase the size of any possible prime divisor of N until it is impossible for N to be the product of two or more primes. These methods usually 2 work as long as $ N < {K^2}$ , where K is $ 1/12$ of the product of the known prime power factors of $ N \pm 1$, $ {N^2} \pm N + 1$, and $ {N^2} + 1$. In this paper a technique is described which, when used in conjunction with these methods, will often determine the pri mality of N when $ N < l{K^3}$ and l is small.


References [Enhancements On Off] (What's this?)

  • [1] JOHN BRILLHART, D. H. LEHMER & J. L. SELFRIDGE, "New primality criteria and factorizations of $ {2^m} \pm 1$," Math. Comp., v. 29, 1975, pp. 620-647. MR 0384673 (52:5546)
  • [2] GARY L. MILLER, "Riemann's hypothesis and tests for primality," J. Comput. System Sci., v. 13, 1976, pp. 300-317. MR 0480295 (58:470a)
  • [3] NOEL B. SLATER, "Gaps and steps for the sequence $ n\theta \bmod\, 1$," Proc. Cambridge Philos. Soc., v. 63, 1967, pp. 1115-1123. MR 0217019 (36:114)
  • [4] R. SOLOVAY & V. STRASSEN, "A fast Monte-Carlo test for primality," SIAM J. Comput., v. 6, 1977, pp. 84-85. MR 0429721 (55:2732)
  • [5] H. C. WILLIAMS & J. S. JUDD, "Determination of the primality of N by using factors of $ {N^2} \pm 1$," Math. Comp., v. 30, 1976, pp. 157-172. MR 0396390 (53:257)
  • [6] H. C. WILLIAMS & J. S. JUDD, "Some algorithms for prime testing using generalized Lehmer functions," Math. Comp., v. 30, 1976, pp. 867-886. MR 0414473 (54:2574)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0476625-0
Article copyright: © Copyright 1978 American Mathematical Society

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