## NOTES ON SOME NEW KINDS OF PSEUDOPRIMES

### BibTeX

@MISC{Zhang_noteson,

author = {Zhenxiang Zhang},

title = {NOTES ON SOME NEW KINDS OF PSEUDOPRIMES},

year = {}

}

### OpenURL

### Abstract

Abstract. J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprime to two bases only, where p = 2 or 3. In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavorable for Browkin’s observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow 2-pseudoprimes to the same bases. In Section 3, we give examples of Sylow p-pseudoprimes to the first several prime bases for the first several primes p. We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow p-pseudoprime to the same bases for all p ∈{2, 3, 5, 7, 11, 13}. In Section 4, we define n to be a k-fold Carmichael Sylow pseudoprime, ifitisaSylowp-pseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n − 1. We find and tabulate all three 3-fold Carmichael Sylow pseudoprimes < 1016. In Section 5, we define a positive odd composite n to be a Sylow uniform pseudoprime to bases b1,...,bk, or a Syl-upsp(b1,...,bk) for short, if it is a Sylp-psp(b1,...,bk) for all the first ω(n − 1) − 1 small prime factors p of n − 1, where ω(n − 1) is the number of distinct prime factors of n − 1. We find and tabulate all the 17 Syl-upsp(2, 3, 5)’s < 1016 and some Syl-upsp(2, 3, 5, 7, 11)’s < 1024. Comparisons of effectiveness of Browkin’s observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6. 1.

### Citations

207 |
Riemann’s hypothesis and tests for primality
- Miller
- 1976
(Show Context)
Citation Context ...lgorithm. This was a theoretical breakthrough, but was pointed out by Günter M. Ziegler [18] that, “it is not yet suitable for use in practice”. A. Stiglic [13] pointed out that the Rabin-Miller test =-=[8, 12]-=- is probably the primality test the most used in practice. Let n>1 be an odd integer and let b1,...,bk be some reduced residues modulo n. Ifnis composite and the congruence (1.1) b n−1 j ≡ 1 mod n Rec... |

72 | There are infinitely many Carmichael numbers
- Alford, Granville, et al.
- 1994
(Show Context)
Citation Context ...ylow p-pseudoprime}. Remark 4.1. It is clear that Browkin’s observation with p odd is not suitable for detecting compositeness of k-fold Carmichael Sylow pseudoprimes. Alford, Granville and Pomerance =-=[2]-=- have proved that there are infinitely many Carmichael numbers. Table 5 suggests that, for any k ≥ 1, there would exist (infinitely many) k-fold Carmichael Sylow pseudoprimes. 5. Sylow uniform pseudop... |

15 | On strong pseudoprimes to several bases
- Jaeschke
(Show Context)
Citation Context ...to many bases which are not Sylow p-pseudoprime to two bases only, where p = 2 or 3. More precisely, in [5, §§4-5] he checked the numbers ψm for 2 ≤ m ≤ 8 and upper bounds of ψ9, ψ10 and ψ11 given in =-=[7]-=- and found that every number of which does not belong to some Sylp-psp(b1,b2) forp = 2 or 3 and b1,b2 ∈ {2, 3, 5}, where ψm is the smallest strong pseudoprime to all the first m prime bases [11]. In [... |

15 |
The pseudoprimes to 25
- Pomerance, Selfridge, et al.
- 1980
(Show Context)
Citation Context ...ven in [7] and found that every number of which does not belong to some Sylp-psp(b1,b2) forp = 2 or 3 and b1,b2 ∈ {2, 3, 5}, where ψm is the smallest strong pseudoprime to all the first m prime bases =-=[11]-=-. In [5, §5] he then verified that for every number n (with one exception) in [7, Table 1: all strong pseudoprimes n<1012 to bases 2, 3 and 5] there exists a prime p ∈{2, 3, 5} and aNOTES ON SOME NEW... |

13 | Efficiency and Security of Cryptosystems Based on Number Theory
- Bleichenbacher
- 1996
(Show Context)
Citation Context ...t) 4185 435 99 25 9 3 1 0 h2(S2,t) 4185 375 88 25 9 3 1 0 h0(S3,t) 52593 52593 4603 606 107 11 2 1 h1(S3,t) 52593 52593 4603 606 107 11 2 1 h2(S3,t) 52593 47614 3866 413 61 10 1 0454 ZHENXIANG ZHANG =-=[3]-=-. We list the values of the functions hi(Sj,t)for0≤ i ≤ 2and1≤ j ≤ 3in Table 1. Remark 2.1. From Table 1 we have h1(Sj,t)=h0(Sj,t)for1≤ j ≤ 3. But there do exist strong pseudoprimes > 10 16 which are ... |

13 |
A course in computational number theory
- Bressoud, Wagon
- 2000
(Show Context)
Citation Context ...-pseudoprimes to the same bases. 3. Sylow p-pseudoprimes to several bases for several p We have checked all the 52593 spsp(2, 3)’s < 1016 given by Bleichenbacher [3] (also available in the package of =-=[4]-=-). In contrast to Browkin’s observation [5, §5] on all the 101 strong pseudoprimes n<1012 to bases 2, 3 and 5, we find all 43 numbers < 1016 which are Sylp-psp(2, 3, 5)’s for all p ∈{2, 3, 5}. Two of ... |

8 |
The Carmichael numbers up to 10
- Pinch
- 1993
(Show Context)
Citation Context ...m2-psp(b1,...,bt), h2(S,t)=#S∩ Syl2-psp(b1,...,bt). Then we have h2(S,t) ≤ h1(S,t) ≤ h0(S,t). Let S1 be the set of 264239 psp(2)’s < 10 13 [10], let S2 be the set of 246683 Carmichael numbers < 10 16 =-=[9]-=-, and let S3 be the set of 52593 spsp(2, 3)’s < 10 16 Table 1. The functions hi(Sj,t) t 1 2 3 4 5 6 7 8 bt 2 3 5 7 11 13 17 19 h0(S1,t) 58892 2696 240 24 2 1 0 0 h1(S1,t) 58892 2696 240 24 2 1 0 0 h2(... |

7 |
Finding strong pseudoprimes to several bases
- Zhang
(Show Context)
Citation Context ... another two of the 43 numbers are Sylp-psp(2, 3, 5)’s for all p ∈{2, 3, 5, 7, 11} listed in Table 3. We have also checked all the 44134 K3-spsp(2, 3, 5, 7, 11)’s < 1024 obtained in our obvious paper =-=[14]-=- and all 330670 K3-spsp(2, 3, 5, 7, 11, 13)’s < 1028 computed recently by us. (We call n = p · q aKk-number [14], if both p and q are primes with q − 1=k(p − 1). A Kk-spsp is a Kk-number and an spsp.)... |

4 | A Computational Perspective - Crandall, Pomerance - 2001 |

1 |
Some new kinds of pseudoprimes
- Browkin
(Show Context)
Citation Context ...eterministic, ERH-based test; see [6, Section 3.4].) The definition of strong pseudoprimes is based on the fact that in a finite field the equation X2 = 1 has at most two solutions, 1 and −1. Browkin =-=[5]-=- defined more general pseudoprimes using the fact that, in a finite field, the equation Xr =1has at most r solutions for every r ≥ 2. Let p be a prime such that n − 1=prm with r>0andp∤m, andlet (1.3) ... |

1 |
The PRIMES is
- Stiglic
(Show Context)
Citation Context ...erministic polynomial time primality proof algorithm. This was a theoretical breakthrough, but was pointed out by Günter M. Ziegler [18] that, “it is not yet suitable for use in practice”. A. Stiglic =-=[13]-=- pointed out that the Rabin-Miller test [8, 12] is probably the primality test the most used in practice. Let n>1 be an odd integer and let b1,...,bk be some reduced residues modulo n. Ifnis composite... |

1 |
The great prime number record races, Notices of the AMS
- Ziegler
(Show Context)
Citation Context ... August 2002, M. Agrawal, N. Kayal and N. Saxena [1] presented a deterministic polynomial time primality proof algorithm. This was a theoretical breakthrough, but was pointed out by Günter M. Ziegler =-=[18]-=- that, “it is not yet suitable for use in practice”. A. Stiglic [13] pointed out that the Rabin-Miller test [8, 12] is probably the primality test the most used in practice. Let n>1 be an odd integer ... |