A one-parameter quadratic-base version of the Baillie-PSW probable prime test

Abstract

The well-known Baillie-PSW probable prime test is a combination of a Rabin-Miller test and a (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 (T2 - uT + 1), and define the base-counting functions: B(n) : #{u: 0 ≤ u < n, n is a psp(Tu)} and SB(n) : #{u: 0 ≤ u < n, n is 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 and passed by an odd composite n = p1r1 P2r2 ...Psrs (P1 < P2 < .... < Ps odd primes) with probability of error τ (n). We give explicit formulas to compute τ(n), and prove that τ(n) < {1/n4/3, for n nonsquare free with s = 1; 1/n2/3, for n square free with s = 2; 1/n2/7, for n square free with s = 3; 1/8s-4.166(p1+1), for n square free with s even ≥ 4; 1/16s-5.119726, for n square free with s odd ≥ 5; 1/4s Πi=1s 1/pi2(ri-1), otherwise, i.e., for n nonsquare free with s ≥ 2. The running time of the OPQBT is asymptotically 4 times that of a Rabin-Miller test for worst cases, but twice that of a Rabin-Miller test for most composites. We point out that the OPQBT has clear finite group (field) structure and nice symmetry, and is indeed a more general and strict version of the Baillie-PSW test. Comparisons with Gantham's RQFT are given.