## TWO KINDS OF STRONG PSEUDOPRIMES UP TO 10 36

### BibTeX

@MISC{Zhang_twokinds,

author = {Zhenxiang Zhang},

title = {TWO KINDS OF STRONG PSEUDOPRIMES UP TO 10 36},

year = {}

}

### OpenURL

### Abstract

Abstract. Let n>1 be an odd composite integer. Write n − 1=2sd with d odd. If either bd ≡ 1modnor b2rd ≡−1modnfor some r =0, 1,...,s − 1, then we say that n isastrongpseudoprimetobaseb, or spsp(b) forshort. Define ψt to be the smallest strong pseudoprime to all the first t prime bases. If we know the exact value of ψt, we will have, for integers n<ψt, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψt are known for 1 ≤ t ≤ 8. Conjectured values of ψ9,...,ψ12 were given by us in our previous papers (Math. Comp. 72 (2003), 2085–2097; 74 (2005), 1009–1024). The main purpose of this paper is to give exact values of ψ ′ t for 13 ≤ t ≤ 19; to give a lower bound of ψ ′ 20: ψ ′ 20> 1036; and to give reasons and numerical evidence of K2- and C3-spsp’s < 1036 to support the following conjecture: ψt = ψ ′ t <ψ′ ′ t for any t ≥ 12, where ψ ′ t (resp. ψ′ ′ t) is the smallest K2- (resp. C3-) strong pseudoprime to all the first t prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp’s < 1036 to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: n = pq with p, q primes and q − 1=2(p−1); and that a C3-spsp is an spsp and a Carmichael number of the form: n = q1q2q3 with each prime factor qi ≡ 3mod4.) 1.

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(Show Context)
Citation Context ... number of the form: n = q1q2q3 with each prime factor qi ≡ 3mod4.) 1. Introduction Let n>1 be an odd integer. Write n − 1=2sd with d odd. We say that n passes the Miller (strong probable prime) test =-=[5]-=- to base b, orthatnis an sprp(b) for short, if (1.1) either b d ≡ 1(modn) orb 2r d ≡−1(modn) forsomer =0, 1,...,s− 1. (The original test of Miller [5] was somewhat more complicated and was a determini... |

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(Show Context)
Citation Context ...by the NSF of China Grant 10071001. c○2007 American Mathematical Society Reverts to public domain 28 years from publication 20952096 ZHENXIANG ZHANG where ϕ is Euler’s function. Monier [6] and Rabin =-=[8]-=- proved that if n is an odd composite positive integer, then SB(n) ≤ (n−1)/4. In fact, as pointed out by Damg˚ard, Landrock and Pomerance [2], if n ̸= 9 is odd and composite, then SB(n) ≤ ϕ(n)/4, i.e.... |

22 |
Prime numbers: A Computational Perspective, 2nd Ed
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(Show Context)
Citation Context ....3) holds with x replaced by d. In [10, Section 4], we speeded up the procedure described in [10, Section 2] so that we can find all C3-spsp’s less than a larger limit L =10 50 ,withthesame signature =-=(1, 37, 41)-=-, to the first t ≥ 9 prime bases. The accelerated procedure is based on the following Lemma 3.2.2102 ZHENXIANG ZHANG Lemma 3.2 ([10, Corollary 4.1]). Let N = q1q2q3 be a product of three different od... |

20 |
Evaluation and comparison of two efficient probabilistic primality testing algorithms
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(Show Context)
Citation Context ...was supported by the NSF of China Grant 10071001. c○2007 American Mathematical Society Reverts to public domain 28 years from publication 20952096 ZHENXIANG ZHANG where ϕ is Euler’s function. Monier =-=[6]-=- and Rabin [8] proved that if n is an odd composite positive integer, then SB(n) ≤ (n−1)/4. In fact, as pointed out by Damg˚ard, Landrock and Pomerance [2], if n ̸= 9 is odd and composite, then SB(n) ... |

15 | On strong pseudoprimes to several bases
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(Show Context)
Citation Context ...have a deterministic primality testing algorithm which is not only easier to implement but also faster than existing deterministic primality testing algorithms. From Pomerance et al. [7] and Jaeschke =-=[4]-=- we know the exact values of ψt for 1 ≤ t ≤ 8 and upper bounds for ψ9, ψ10 and ψ11: ψ9 ≤ 41234 31613 57056 89041 (20 digits) = 4540612081 · 9081224161, ψ10 ≤ 155 33605 66073 14320 55410 02401 (28 digi... |

15 |
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(Show Context)
Citation Context ...ers n<ψt we will have a deterministic primality testing algorithm which is not only easier to implement but also faster than existing deterministic primality testing algorithms. From Pomerance et al. =-=[7]-=- and Jaeschke [4] we know the exact values of ψt for 1 ≤ t ≤ 8 and upper bounds for ψ9, ψ10 and ψ11: ψ9 ≤ 41234 31613 57056 89041 (20 digits) = 4540612081 · 9081224161, ψ10 ≤ 155 33605 66073 14320 554... |

7 |
Finding strong pseudoprimes to several bases
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(Show Context)
Citation Context ... sense that: (1.4) n = q1q2q3 with q1 <q2 <q3 odd primes and each qi − 1 | n − 1. For short we call numbers (strong pseudoprimes) having the form (1.3) Kk-numbers (spsp’s), say, K2-spsp’s if k =2. In =-=[9]-=-, we used biquadratic residue characters and cubic residue characters as main tools to find all K2-, K3-, K4-strong pseudoprimes < 1024 to the first nine or ten prime bases. As a result the upper boun... |

3 |
Average case estimates for the strong probable prime test
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(Show Context)
Citation Context ...NG ZHANG where ϕ is Euler’s function. Monier [6] and Rabin [8] proved that if n is an odd composite positive integer, then SB(n) ≤ (n−1)/4. In fact, as pointed out by Damg˚ard, Landrock and Pomerance =-=[2]-=-, if n ̸= 9 is odd and composite, then SB(n) ≤ ϕ(n)/4, i.e., PR(n) ≤ 1/4. These facts lead to the Rabin-Miller test: given a positive integer n, pickkdifferent positive integers less than n and perfor... |