A note on Perrin pseudoprimes

Arno, Steven

doi:10.1090/S0025-5718-1991-1052083-9

A note on Perrin pseudoprimes
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by Steven Arno PDF

Math. Comp. 56 (1991), 371-376 Request permission

Abstract:

The cubic recurrence $A(n + 3) = A(n) + A(n + 1)$ with initial conditions $A(0) = 3$, $A(1) = 0$, $A(2) = 2$, known as Perrin’s sequence, is associated with several types of pseudoprimes. In this paper we will explore a question of Adams and Shanks concerning the existence of the so-called Q and I Perrin pseudoprimes, and develop an algorithm to search for all such pseudoprimes below some specified limit. As an example, we show that none exist below ${10^{14}}$.

References

William Adams and Daniel Shanks, Strong primality tests that are not sufficient, Math. Comp. 39 (1982), no. 159, 255–300. MR 658231, DOI 10.1090/S0025-5718-1982-0658231-9
William W. Adams, Characterizing pseudoprimes for third-order linear recurrences, Math. Comp. 48 (1987), no. 177, 1–15. MR 866094, DOI 10.1090/S0025-5718-1987-0866094-6
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234

Sur la recherche de grands nombres premiers

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G. C. Kurtz, Daniel Shanks, and H. C. Williams, Fast primality tests for numbers less than $50\cdot 10^9$, Math. Comp. 46 (1986), no. 174, 691–701. MR 829639, DOI 10.1090/S0025-5718-1986-0829639-7