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**1 - 2**of**2**### Pseudoprimes: A Survey Of Recent Results

, 1992

"... this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers. First of all, we make a list of all pseudoprime varieties existing so far. This includes Lucas-pseudoprimes and the generalization to sequences generated by integer polynomials modulo N , elliptic ..."

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this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers. First of all, we make a list of all pseudoprime varieties existing so far. This includes Lucas-pseudoprimes and the generalization to sequences generated by integer polynomials modulo N , elliptic pseudoprimes. We discuss the making of tables and the consequences on the design of very fast primality algorithms for small numbers. Then, we describe the recent work of Alford, Granville and Pomerance, in which they prove that there

### THERE ARE INFINITELY MANY PERRIN PSEUDOPRIMES

"... Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zero-den ..."

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Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zero-density estimates for Hecke L-functions. 1. Background In a 1982 paper [1], Adams and Shanks introduced a probable primality test based on third order recurrence sequences. The following is a version of that test. Consider sequences An = An(r, s) defined by the following relations: A−1 = s, A0 = 3, A1 = r, and An = rAn−1 − sAn−2 + An−3. Let f(x) = x 3 − rx 2 + sx − 1 be the associated polynomial and ∆ its discriminant. (Perrin’s sequence is An(0, −1).) Definition. The signature mod m of an integer n with respect to the sequence Ak(r, s) is the 6-tuple (A−n−1, A−n, A−n+1, An−1, An, An+1) mod m. Definitions. An integer n is said to have an S-signature if its signature mod n is congruent to (A−2, A−1, A0, A0, A1, A2). An integer n is said to have a Q-signature if its signature mod n is congruent to (A, s, B, B, r, C), where for some integer a with f(a) ≡ 0 mod n, A ≡ a −2 + 2a, B ≡ −ra 2 + (r 2 − s)a, and C ≡ a 2 + 2a −1. An integer n is said to have an I-signature if its signature mod n is congruent to (r, s, D ′ , D, r, s), where D ′ + D ≡ rs − 3 mod n and (D ′ − D) 2 ≡ ∆. Definition. A Perrin pseudoprime with parameters (r, s) is an odd composite n such that either