Strong primality tests that are not sufficient

Authors:
William Adams and Daniel Shanks

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

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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 by

*n*is prime, . Perrin asked if any composite satisfies this congruence if , . The answer is yes, and our first example leads us to strengthen the condition by introducing the "signature" of

*n*:

*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 ths of all primes. is easily computable in 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.

**[1]**R. Perrin, "Item 1484,"*L'Intermédiare des Math.*, v. 6, 1899, pp. 76-77.**[2]**E. Malo,*ibid.*, v. 7, 1900, p. 281, p. 312.**[2a]**E. B. Escott,*ibid.*, v. 8, 1901, pp. 63-64.**[3]**Dov Jarden,*Recurring Sequences*, Riveon Lematematika, Jerusalem, 1966.**[4]**Daniel Shanks,*Calculation and applications of Epstein zeta functions*, Math. Comp.**29**(1975), 271–287. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR**0409357**, https://doi.org/10.1090/S0025-5718-1975-0409357-2

Daniel Shanks,*Corrigenda: “Calculation and applications of Epstein zeta functions” (Math. Comp. 29 (1975), 271–287)*, Math. Comp.**29**(1975), no. 132, 1167. MR**0409358**, https://doi.org/10.1090/S0025-5718-1975-0409358-4

Daniel Shanks,*Corrigendum: “Calculation and applications of Epstein zeta functions” (Math. Comp. 29 (1975), 271–287)*, Math. Comp.**30**(1976), no. 136, 900. MR**0409359**, https://doi.org/10.1090/S0025-5718-1976-0409359-7**[5]**Daniel Shanks,*Five number-theoretic algorithms*, Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. Congressus Numerantium, No. VII. MR**0371855****[6]**Donald Ervin Knuth,*Seminumerical Algorithms*, Second printing, Addison-Wesley, Reading, Mass., 1971, esp. pp. 260-266.**[7]**Carl Pomerance, J. L. Selfridge, and Samuel S. Wagstaff Jr.,*The pseudoprimes to 25⋅10⁹*, Math. Comp.**35**(1980), no. 151, 1003–1026. MR**572872**, https://doi.org/10.1090/S0025-5718-1980-0572872-7**[8]**Daniel Shanks, "Review of Fröberg,"*ibid.*, v. 29, 1975, pp. 331-333.**[9]**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), Utilitas Math., Winnipeg, Man., 1976, pp. 15–40. Congressus Numerantium, No. XVII. MR**0453691****[10]**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****[11]**Daniel Shanks,*Solved and unsolved problems in number theory*, 2nd ed., Chelsea Publishing Co., New York, 1978. MR**516658****[12]**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**0384744**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658231-9

Article copyright:
© Copyright 1982
American Mathematical Society