## On the Distributions of Pseudoprimes, Carmichael Numbers, and (2009)

### BibTeX

@MISC{Pseudoprimes09onthe,

author = {Strong Pseudoprimes and Aran Nayebi},

title = {On the Distributions of Pseudoprimes, Carmichael Numbers, and},

year = {2009}

}

### OpenURL

### Abstract

Building upon the work of Carl Pomerance and others, the central purpose of this discourse is to discuss the distribution of base-2 pseudoprimes, as well as improve upon Pomerance's conjecture regarding the Carmichael number counting function [8]. All conjectured formulas apply to any base b ≥ 2 for x ≥ x0(b). A table of base-2 pseudoprime, 2-strong pseudoprime, and Carmichael number counts up to 10 15 from [4] is included in the Appendix. We also discuss strong pseudoprimes and probabilistic primality testing. 1

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(Show Context)
Citation Context ...oted by πk(x) and let the number of composites ≤ x with k ≥ 2 prime factors (not necessarily distinct) be represented by τk(x). Hence, we can prove upper and lower bounds for πk(x). Due to 22.18.2 in =-=[7]-=- for k ≥ 1, where Πk(x) = ∑ 1 = ϑk(x) log x Since ϑk(x) = Πk(x) log x − ∫ x 2 Πk(x) dt = O(x), Πk(x) ∼ t 2 k!πk(x) ≤ Πk(x) ≤ k!τk(x), (2.2.17) x + O( log x ) by 22.18.5, such that ϑk(x) = ∑ log(p1p2· ... |

243 |
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Citation Context ... Poisson distribution P(k; λ) and taking its limit for any xed k, (log log x) lim P(k; λ) = lim x→∞ x→∞ k−1 exp{− log log x} (k − 1)! = 0, (2.2.22) 1 where the asymptotic error bound is given by O( ) =-=[13]-=-. However, just because the log log x probability of a general composite near x having k distinct prime factors goes to 0, does not necessitate that this probability will hold for either P πbkγ (x) or... |

145 |
Prime Numbers: a Computational Perspective
- Crandall, Pomerance
(Show Context)
Citation Context ...n can only ensure that n is probably prime for there are odd composite integers called pseudoprimes to an arbitrary base b ≥ 2 which satisfy (1.1.2). 1.2 Previous Results and Derivations Erd®s proved =-=[1]-=- that for the pseudoprimes de ned in (1.1.2), Lemma 1 (Erd®s 1950). lim P πb(x) = o(π(x)), where P πb(x) is the number of base-b x→∞ pseudoprimes ≤ x for b ≥ 2. Thus, the number of pseudoprimes is alw... |

22 |
On the distribution of pseudoprimes
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(Show Context)
Citation Context ...mes ≤ x for b ≥ 2. Thus, the number of pseudoprimes is always less than the number of primes. Although the claim established in Lemma 1 is rather elementary, Erd®s went on to formulate an upper-bound =-=[11]-=- for the pseudoprime counting function. 1Lemma 2 (Lehmer 1936, Erd®s 1956). For the constants ɛ1 > 01 and ɛ2 < 0, ɛ1log x2 √ ≤ P π2(x) ≤ x · exp{ɛ2 log x log log x}. Not long afterwards, Pomerance im... |

14 | Two contradictory conjectures concerning Carmichael numbers
- Granville, Pomerance
(Show Context)
Citation Context ...d others, the central purpose of this discourse is to discuss the distribution of base-2 pseudoprimes, as well as improve upon Pomerance's conjecture regarding the Carmichael number counting function =-=[8]-=-. All conjectured formulas apply to any base b ≥ 2 for x ≥ x0(b). A table of base-2 pseudoprime, 2-strong pseudoprime, and Carmichael number counts up to 10 15 from [4] is included in the Appendix. We... |

9 | The distribution of Lucas and elliptic pseudoprimes
- Gordon, Pomerance
- 1991
(Show Context)
Citation Context ...ction. 1Lemma 2 (Lehmer 1936, Erd®s 1956). For the constants ɛ1 > 01 and ɛ2 < 0, ɛ1log x2 √ ≤ P π2(x) ≤ x · exp{ɛ2 log x log log x}. Not long afterwards, Pomerance improved these bounds signi cantly =-=[6]-=- [11], Lemma 3 (R.A. Mollin ed. 1989, Pomerance 1981). For the base-2 pseudoprime counting function, exp{(log x) 85 207 } 3≤ P π2(x) ≤ x · L(x) −1 log x log log log x 2 , where L(x) = exp{ }. These lo... |

5 |
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(Show Context)
Citation Context ... erent because pi > xi due to the fact that xi is composite and pi ≡ 1 (mod (xi − 1)). Hence, Lemma 4 becomes apparent. 1 According to [10], ɛ1 = 5 8 log 2 . 2 The lower bound is accredited to Lehmer =-=[9]-=-. 3 Pomerance proved that P π2(x) ≥ exp{(log x) 5 14 }[9], but Mollin improved this lower bound to P π2(x) ≥ exp{(log x) 85 207 }. 2Lemma 5. In a similar manner to Lemma 4, we extend Szymiczek's proo... |

3 |
Evaluation and Comparison of two e cient probabilistic primality testing algorithms
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Citation Context ...dd and n − 1 = 2 s k if either b k ≡ 1 (mod n) (4.0.5) or for j < s. Next, let b 2j k ≡ −1 (mod n) (4.0.6) S(n) := #{b ∈ [1, n − 1] : b k ≡ 1 (mod n) ∨ b 2j k ≡ −1 (mod n) for some j < s}. (4.0.7) In =-=[17]-=-, Monier proved, Lemma 16. Take the r to be the largest odd factor of n−1 and let v(n) be the largest number for which 2v(n) | p − 1 for p | n, where p is a prime. Thus, ⎛ v(n)−1 ∑ S(n) = ⎝2 + 2 j·ω(n... |

2 |
On the Converse of Fermat's Theorem
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- 1949
(Show Context)
Citation Context ...mes for su ciently large x ≥ x0(b) is [11], P π2(x) ∼ x · L(x) −1 . (1.2.2) 2 Preliminaries 2.1 Two Lemmas Regarding Base-b Pseudoprimes with kγ Distinct Prime Factors Lemma 4. We extend Erd®s' proof =-=[2]-=- to prove that there exist in nitely many square-free log x base-b pseudoprimes for b ≥ 2 with 2 ≤ α(x) = kγ << distinct prime factors following log log x extremely closely from the case for base-b de... |

2 |
Tables of pseudoprimes and related data
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(Show Context)
Citation Context ...ichael number counting function [8]. All conjectured formulas apply to any base b ≥ 2 for x ≥ x0(b). A table of base-2 pseudoprime, 2-strong pseudoprime, and Carmichael number counts up to 10 15 from =-=[4]-=- is included in the Appendix. We also discuss strong pseudoprimes and probabilistic primality testing. 1 Introduction 1.1 Background Information Fermat's little is prime if theorem, also known as Ferm... |

2 | The Pseudoprimes below 2 64
- Galway
- 2002
(Show Context)
Citation Context ...(r − s)) (rs) 3 2 ≈ 30.03, (3.1.2) α = 2 ∏ 1 − 2/d ≈ 1.320323632, (3.1.3) (1 − 1/d) 2 d ρ(m) = ∏d − 1 , (3.1.4) d − 2 d|m ⎧ ⎪⎨ 2, if 4 | m δ(m) = (3.1.5) ⎪⎩ 1, if otherwise According to his data from =-=[5]-=-, Galway's conjecture is somewhat supported: Bound P2(x) ζ2 104 11 9.331 105 34 14.251 106 107 20.423 107 311 25.550 108 880 29.860 109 2455 33.340 10 10 6501 34.468 1011 17207 34.908 1012 46080 35.18... |

2 |
On Pseudoprimes which are Products of Distinct Primes
- Szymiczek
- 1967
(Show Context)
Citation Context ...3 Pomerance proved that P π2(x) ≥ exp{(log x) 5 14 }[9], but Mollin improved this lower bound to P π2(x) ≥ exp{(log x) 85 207 }. 2Lemma 5. In a similar manner to Lemma 4, we extend Szymiczek's proof =-=[12]-=- to prove that for any base-b pseudoprime, b ≥ 2, having kγ ≥ 2 distinct prime factors and for x su ciently large, P πbkγ +1(x) ≥ P πbkγ (logb x). (2.1.1) log x log log x Proof. Let n be a pseudoprime... |

2 | A One-Parameter Quadratic-Base Version of the Baillie-PSW
- Zhang
- 2002
(Show Context)
Citation Context ...imality testing. However, as demonstrated, the latter criteria are not su cient by today's standards. A recent test known as the One-Parameter Quadratic-Base Test (OPQBT) presented by Zhenxiang Zhang =-=[14]-=- reduces the margin of error signi cantly. Lemma 19 (OPQBT). If we let OP QBT E(n) denote the probability that the same odd composite n = p1 β1 p2 β2 · · · pk βk passes the One-Parameter Quadratic-Bas... |

2 |
A Probable Prime Test with High Con dence. Journal of Number Theory 72
- Grantham
- 1998
(Show Context)
Citation Context ...d k ≥ 5 119726 ⎪⎩ 4−k k∏ pγ 2(1−rγ) , if n is nonsquare free with k ≥ 2 γ=1 We propose a rather trivial alternative, that is, to combine the Randomized Quadratic Frobenius Test (RQFT) of Jon Grantham =-=[15]-=- with the OPQBT test. Thus, 1 iteration in this proposed algorithm would run the RQFT test and if n is not determined composite, a subsequent OPQBT test will be run. In its worst case, the algorithm s... |