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New primality criteria and factorizations of $2^{m}\pm 1$

Authors: John Brillhart, D. H. Lehmer and J. L. Selfridge
Journal: Math. Comp. 29 (1975), 620-647
MSC: Primary 10A25
Erratum: Math. Comp. 39 (1982), 747-757.
Erratum: Math. Comp. 39 (1982), 747.
MathSciNet review: 0384673
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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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Keywords: Primality testing, factorization of <!– MATH ${2^m} \pm 1$ –> <IMG WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${2^m} \pm 1$">, Lucas sequences, converse of Fermat’s theorem
Article copyright: © Copyright 1975 American Mathematical Society