Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Determination of the primality of $ N$ by using factors of $ N\sp{2}\pm 1$


Authors: H. C. Williams and J. S. Judd
Journal: Math. Comp. 30 (1976), 157-172
MSC: Primary 10A25
DOI: https://doi.org/10.1090/S0025-5718-1976-0396390-3
MathSciNet review: 0396390
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Algorithms are developed which can be used to determine the primality of a large integer N when a sufficient number of prime factors of $ {N^2} + 1$ are known. A test for the primality of N which makes use of known factors of $ N - 1,N + 1$ and $ {N^2} + 1$ and the factor bounds on these numbers is also presented. In order to develop the necessary theory, the properties of some functions which are a generalization of Lehmer functions are used. Several examples of numbers proved prime by employing these tests are given.


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] DOV JARDEN, Recurring Sequences, 3rd ed., Riveon Lemathematika, Jerusalem, 1973, pp. 41-59.
  • [3] D. H. LEHMER, "Computer technology applied to the theory of numbers," Studies in Number Theory, Math. Assoc. Amer.; distributed by Prentice-Hall, Englewood Cliffs, N. J., 1969, pp. 117-151. MR 40 #84. MR 0246815 (40:84)
  • [4] M. A. MORRISON, "A note on primality testing using Lucas sequences," Math. Comp., v. 29, 1975, pp. 181-182. MR 0369234 (51:5469)
  • [5] J. M. POLLARD, "Theorems on factorization and primality testing," Proc. Cambridge Philos. Soc., v. 76, 1974, pp. 521-528. MR 0354514 (50:6992)
  • [6] J. L. SELFRIDGE & M. C. WUNDERLICH, "An efficient algorithm for testing large numbers for primality," Proc. Fourth Manitoba Conf. on Numerical Math., Winnipeg, Manitoba, 1974, pp. 109-120. MR 0369226 (51:5461)
  • [7] H. C. WILLIAMS, "A generalization of Lehmer's functions," Acta Arith. (To appear).

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10A25

Retrieve articles in all journals with MSC: 10A25


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1976-0396390-3
Keywords: Primality testing, Lucas functions, Lehmer functions
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society