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Strong primality tests that are not sufficient
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- by William Adams and Daniel Shanks PDF
- Math. Comp. 39 (1982), 255-300 Request permission
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, $A(n) \equiv A(1)\;\pmod n$. 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: \[ A( - n - 1),A( - n),A( - n + 1),A(n - 1),A(n),A(n + 1)\] $\bmod n$. Primes have three types of signatures depending on how they split in the cubic field generated by ${x^3} - r{x^2} + sx - 1 = 0$. 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 $5/6$ths of all primes. $A(n)\;\pmod n$ 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.References
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R. Perrin, "Item 1484," L’Intermédiare des Math., v. 6, 1899, pp. 76-77.
E. Malo, ibid., v. 7, 1900, p. 281, p. 312.
E. B. Escott, ibid., v. 8, 1901, pp. 63-64.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966.
- Daniel Shanks, Calculation and applications of Epstein zeta functions, Math. Comp. 29 (1975), 271–287. MR 409357, DOI 10.1090/S0025-5718-1975-0409357-2
- Daniel Shanks, Five number-theoretic algorithms, Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Congressus Numerantium, No. VII, Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. MR 0371855 Donald Ervin Knuth, Seminumerical Algorithms, Second printing, Addison-Wesley, Reading, Mass., 1971, esp. pp. 260-266.
- Carl Pomerance, J. L. Selfridge, and Samuel S. Wagstaff Jr., The pseudoprimes to $25\cdot 10^{9}$, Math. Comp. 35 (1980), no. 151, 1003–1026. MR 572872, DOI 10.1090/S0025-5718-1980-0572872-7 Daniel Shanks, "Review of Fröberg," ibid., v. 29, 1975, pp. 331-333.
- Daniel Shanks, A survey of quadratic, cubic and quartic algebraic number fields (from a computational point of view), Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Congressus Numerantium, No. XVII, Utilitas Math., Winnipeg, Man., 1976, pp. 15–40. MR 0453691
- Daniel Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR 0316385
- Daniel Shanks, Solved and unsolved problems in number theory, 2nd ed., Chelsea Publishing Co., New York, 1978. MR 516658
- William G. Spohn Jr., Letter to the editor: “Incredible identities” (Fibonacci Quart. 12 (1974), 271, 280) by Daniel Shanks, Fibonacci Quart. 14 (1976), no. 1, 12. MR 384744
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp. 39 (1982), 255-300
- MSC: Primary 10A25; Secondary 10-04, 12-04
- DOI: https://doi.org/10.1090/S0025-5718-1982-0658231-9
- MathSciNet review: 658231