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Strong Primality Tests that are Not Sufficient

William Adams and Daniel Shanks
Mathematics of Computation
Vol. 39, No. 159 (Jul., 1982), pp. 255-300
DOI: 10.2307/2007637
Stable URL: http://www.jstor.org/stable/2007637
Page Count: 46
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Strong Primality Tests that are Not Sufficient
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Abstract

A detailed investigation is given of the possible use of cubic recurrences in primality tests. No attempt is made in this abstract to cover all of the many topics examined in the paper. Define a doubly infinite set of sequences $A(n)$ by $$A(n + 3) = rA(n + 2) - sA(n + 1) + A(n)$$ with $A(-1) = s$, $A(0) = 3$, and $A(1) = r$. If $n$ is prime, $A(n) \equiv A(1) (\operatorname{mod} n)$. Perrin asked if any composite satisfies this congruence if $r = 0, s = -1$. The answer is yes, and our first example leads us to strengthen the condition by introducing the "signature" of $n$: $$A(-n - 1), A(-n), A(-n + 1), A(n - 1), A(n), A(n + 1)$$ $\operatorname{mod} n$. Primes have three types of signatures depending on how they split in the cubic field generated by $x^3 - rx^2 + sx - 1 = 0$. Composites with "acceptable" signatures do exist but are very rare. The $S$-type signature, which corresponds to the completely split primes, has a very special role, and it may even be that $I$ and $Q$ type composites do not occur in Perrin's sequence even though the $I$ and $Q$ primes comprise 5/6ths of all primes. $A(n) (\operatorname{mod} n)$ is easily computable in $O(\log n)$ operations. The paper closes with a $p$-adic analysis. This powerful tool sets the stage for our [12] which will be Part II of the paper.

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