Abstract

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

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