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Journal Article

# New Primality Criteria and Factorizations of $2^m \pm 1$

John Brillhart, D. H. Lehmer and J. L. Selfridge
Mathematics of Computation
Vol. 29, No. 130 (Apr., 1975), pp. 620-647
DOI: 10.2307/2005583
Stable URL: http://www.jstor.org/stable/2005583
Page Count: 28

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## Abstract

A collection of theorems is developed for testing a given integer $N$ for primality. The first type of theorem considered is based on the converse of Fermat's theorem and uses factors of $N - 1$. The second type is based on divisibility properties of Lucas sequences and uses factors of $N + 1$. The third type uses factors of both $N - 1$ and $N + 1$ and provides a more effective, yet more complicated, primality test. The search bound for factors of $N \pm 1$ and properties of the hyperbola $N = x^2 - y^2$ are utilized in the theory for the first time. A collection of 133 new complete factorizations of $2^m \pm 1$ and associated numbers is included, along with two status lists: one for the complete factorizations of $2^m \pm 1$; the other for the original Mersenne numbers.

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