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

Venue: | Math. Comp |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Zhang_aone-parameter,

author = {Zhenxiang Zhang},

title = {A one-parameter quadratic-base version of the Baillie–PSW probable prime test},

journal = {Math. Comp},

year = {},

pages = {2002}

}

### OpenURL

### 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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Citation Context ..., then we say that n passes the Rabin-Miller (strong probable prime) test [15] to base b; if in addition, n is composite, then we say n is a strong pseudoprime to base b, or spsp(b) for short. Monier =-=[16]-=- gave a formula for counting the number of bases b such that n is an spsp(b). Both Rabin [20] and Monier [16] proved that if n is an odd composite positive integer, then n passes the Rabin-Miller test... |

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Citation Context ...re we write n −(D/n)=2 k q with q odd. If n is composite and relatively prime to 2QD such that condition (1.4) or (1.5) holds, then we call n a Lucas pseudoprime [4, 19] or a strong Lucas pseudoprime =-=[3, 4]-=- to parameters P and Q, or lpsp(P,Q) or slpsp(P,Q) for short. Let D be an integer, and n a composite number relatively prime to 2D and distinct from 9. Arnault [3] gave a formula to compute the base-c... |

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Citation Context ...s more complicated analysis and computation. But it cannot be improved to n2/3 , e.g., n = 62164241 = 41 · 881 · 1721, B(n,1) = 636519, B(n, −1) = 176000, 18.19 ··· τ0(n)= n2/3 . Remark 5.2. Williams =-=[22]-=- asked whether there are any Carmichael numbers n with an odd number of prime divisors and the additional property that for p | n, p +1 | n + 1. Lemma 5.4 shows that if such a Carmichael number exists... |

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Citation Context ...s then we call n a pseudoprime to base b,orpsp(b) for short. There are composite integers, called Carmichael numbers, such that (1.1) holds for every b with gcd(n,b)=1.Alford, Granville and Pomerance =-=[1]-=- proved that there are infinitely many Carmichael numbers. If (1.2) holds, then we say that n passes the Rabin-Miller (strong probable prime) test [15] to base b; if in addition, n is composite, then ... |

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Citation Context ...T]/(T 2 −uT +1) was first used by the author for factoring large integers near group orders [24]. The idea of using this ring in primality testing is motivated from the Lucas-Lehmer Test described in =-=[7, 8]-=-, where the ring Z[T]/(T 2 − uT − 1) was used. Lenstra’s Galois Theory Test [14] is a method of proving primality using finite fields. 2. Definitions and main results Let u (�= ±2) ∈ Z. PutTu = T mod ... |

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Citation Context ...th Grantham’s RQFT are given in Section 8. Brief conclusions are given in Section 9. Remark 1.1. The ring Z[T]/(T 2 −uT +1) was first used by the author for factoring large integers near group orders =-=[24]-=-. The idea of using this ring in primality testing is motivated from the Lucas-Lehmer Test described in [7, 8], where the ring Z[T]/(T 2 − uT − 1) was used. Lenstra’s Galois Theory Test [14] is a meth... |