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Fast primality tests for numbers less than $ 50\cdot 10\sp 9$

Authors: G. C. Kurtz, Daniel Shanks and H. C. Williams
Journal: Math. Comp. 46 (1986), 691-701
MSC: Primary 11Y11; Secondary 11A51, 11R16
MathSciNet review: 829639
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Abstract: Consider the doubly infinite set of sequences $ A(n)$ given by

$\displaystyle A(n + 3) = rA(n + 2) - sA(n + 1) + A(n)$

with $ A( - 1) = s$, $ A(0) = 3$, $ A(1) = r$. For a given pair r,s, the "signature" of n is defined to be the sextet

$\displaystyle A( - n - 1),A( - n),A( - n + 1),A(n - 1),A(n),A(n + 1),$

each reduced modulo n. Primes have only three types of signatures, depending on how they split in the cubic field generated by $ {x^3} - r{x^2} + sx - 1$. An "acceptable" composite is a composite integer which has the same type of signature as a prime; such integers are very rare. In this paper, a description is given of the results of a computer search for all acceptable composites $ \leqslant 50 \cdot {10^9}$ in the Perrin sequence $ (r = 0,s = - 1)$. Also, some numbers which are acceptable composites for both the Perrin sequence and the sequence with $ r = 1$, $ s = 0$ are presented.

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Article copyright: © Copyright 1986 American Mathematical Society