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

Authors: H. C. Williams and J. S. Judd
Journal: Math. Comp. 30 (1976), 157-172
MSC: Primary 10A25
MathSciNet review: 0396390
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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.

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Keywords: Primality testing, Lucas functions, Lehmer functions
Article copyright: © Copyright 1976 American Mathematical Society