. 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 +...

Abstract. The well-known Baillie-PSW probable prime test is a combination of a Rabin-Miller test and a “true ” (i.e., with (D/n) =−1) Lucas test. Arnault mentioned in a recent paper that no precise result is known about its probability of error. Grantham recently provided a probable prime test (RQFT) with probability of error less than 1/7710, and pointed out that the lack of counter-examples to the Baillie-PSW test indicates that the true probability of error may be much lower. In this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter: Tu = T mod (T 2 − uT + 1), and define the base-counting functions: B(n) =#{u:0 ≤ u<n, nis a psp(Tu)} and SB(n) =#{u:0 ≤ u<n, nis an spsp(Tu)}. Then we give explicit formulas to compute B(n) and SB(n), and prove that, for odd composites n, B(n) <n/2 and SB(n) <n/8, and point out that these are best possible. Finally, based on one-parameter quadratic-base pseudoprimes, we provide a probable prime test, called the One-Parameter Quadratic-Base Test (OPQBT), which passed by all primes ≥ 5 andpassedbyanoddcompositen = p r1 1 pr2 2 ···prs s (p1 <p2 < ·· · <ps odd primes) with probability of error τ(n). We give explicit formulas to compute τ(n), and prove that

Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1.