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A Probable Prime Test With High Confidence
"... . Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automorphism. The test, along with a fixed number of trial divisions, ensure ..."
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. Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automorphism. The test, along with a fixed number of trial divisions, ensures that a composite n will pass for less than 1 7710 of the polynomials x 2 \Gamma bx \Gamma c with i b 2 +4c n j = \Gamma1 and \Gamma \Gammac n \Delta = 1. The running time of the test is asymptotically 3 times that of the Strong Probable Prime Test. x1 Background Perhaps the most common method for determining whether or not a number is prime is the Strong Probable Prime Test. Given an odd integer n, let n = 2 r s + 1 with s odd. Choose a random integer a with 1 a n \Gamma 1. If a s j 1 mod n or a 2 j s j \Gamma1 mod n for some 0 j r \Gamma 1, then n passes the test. An odd prime will pass the test for all a. The test is very fast; it requires no more than (1 +...
MATHEMATICS OF COMPUTATION
, 2000
"... Abstract. We define a Carmichael number of order m to be a composite integer n such that nthpower raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZmodule by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, an ..."
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Abstract. We define a Carmichael number of order m to be a composite integer n such that nthpower raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZmodule by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indicates that for every m there should be infinitely many Carmichael numbers of order m. The argument suggests a method for finding examples of higherorder Carmichael numbers; we use the method to provide examples of Carmichael numbers of order 2. 1.
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 zeroden ..."
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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 zerodensity estimates for Hecke Lfunctions. 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 6tuple (A−n−1, A−n, A−n+1, An−1, An, An+1) mod m. Definitions. An integer n is said to have an Ssignature if its signature mod n is congruent to (A−2, A−1, A0, A0, A1, A2). An integer n is said to have a Qsignature 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 Isignature 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
LUCAS SEQUENCES {Uk} FOR WHICH U2p AND Up ARE PSEUDOPRIMES FOR ALMOST ALL PRIMES p
, 2003
"... It was proven by Emma Lehmer that for almost all odd primes p, F2p is a Fibonacci pseudoprime. In this paper, we generalize this result to Lucas sequences {Uk}. In particular, we find Lucas sequences {Uk} for which either U2p is a Lucas pseudoprime for almost all odd primes p or Up is a Lucas pseudo ..."
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It was proven by Emma Lehmer that for almost all odd primes p, F2p is a Fibonacci pseudoprime. In this paper, we generalize this result to Lucas sequences {Uk}. In particular, we find Lucas sequences {Uk} for which either U2p is a Lucas pseudoprime for almost all odd primes p or Up is a Lucas pseudoprime for almost all odd primes p. 1.
ON THE NUMBER OF PRIME DIVISORS OF
, 2001
"... The classical Carmichael numbers are well known in number theory. These numbers were introduced independently by Korselt in [8] and Carmichael in [2] and since then they have been the subject of intensive study. The reader may find extensive but not exhaustive lists of references in [5, Sect. A13], ..."
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The classical Carmichael numbers are well known in number theory. These numbers were introduced independently by Korselt in [8] and Carmichael in [2] and since then they have been the subject of intensive study. The reader may find extensive but not exhaustive lists of references in [5, Sect. A13], [11, Ch. 2, Sec. IX].