## On the Distribution of Carmichael Numbers (906)

### BibTeX

@MISC{Nayebi906onthe,

author = {Aran Nayebi},

title = {On the Distribution of Carmichael Numbers},

year = {906}

}

### OpenURL

### Abstract

### Citations

939 |
An introduction to the theory of numbers
- Hardy, Wright
- 1979
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Citation Context ...k(x) and let the number of composites ≤ x with k prime factors (not necessarily distinct) be represented by τk(x). Hence, we can prove upper and lower bounds for πk(x). In 22.18.2 of Hardy and Wright =-=[7]-=- for k ≥ 1, k!πk(x) ≤ Πk(x) ≤ k!τk(x), (3.3.2) where Πk(x) = ϑk(x) x +O( log x log x ) in 22.18.5. In 22.18.24, since ϑk(x) = Πk(x) log x− kx(log log x) k−1 for k ≥ 2 and that ∫ x 2 Πk(x) dt = O(x), Π... |

243 |
Introduction to analytic and probabilistic number theory
- Tenenbaum
- 1995
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Citation Context ...e 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, (3.3.7) 1 where the asymptotic error bound is given by O( ) =-=[13]-=-. However, we caution the reader log log x to consider that just because the probability of a general composite near x having k distinct prime factors goes to 0, does not necessitate that this probabi... |

145 |
Prime Numbers: a Computational Perspective
- Crandall, Pomerance
(Show Context)
Citation Context ...ore delving into the main results of this paper, we shall we will explicitly use later on in our derivations. rst present previous results that 22.1 Pseudoprimes 2.1.1 Pseudoprime Distribution Erd®s =-=[1]-=- proved that for the pseudoprimes de ned in (1.0.2), Theorem 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 pseudoprimes are rar... |

22 |
On the distribution of pseudoprimes
- Pomerance
- 1981
(Show Context)
Citation Context ...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 pseudoprimes are rarer than the primes. Although this claim is rather elementary, Erd®s =-=[11]-=- went on to formulate an upper-bound for the pseudoprime counting function. Theorem 2 (Lehmer 1936, Erd®s 1956). For some constants ɛ1 1 and ɛ2, ɛ1log x2 √ ≤ P π2(x) ≤ x · exp{ɛ2 log x log log x}. In ... |

14 | Two contradictory conjectures concerning Carmichael numbers
- Granville, Pomerance
(Show Context)
Citation Context ...ality test, and thus their distribution will be the interest of this paper. Let Pb(x) denote the number of base b pseudoprimes ≤ x; and let C(x) denote the number of Carmichael numbers ≤ x. Pomerance =-=[8]-=- conjectures that for all large x, {1+o(1)} log log log x 1− C(x) = x log log x . (1.0.3) Unfortunately, according to Richard Pinch [19], Pomerance's conjecture does not appear to be well-supported by... |

9 | The distribution of Lucas and elliptic pseudoprimes
- Gordon, Pomerance
- 1991
(Show Context)
Citation Context ...ulate an upper-bound for the pseudoprime counting function. Theorem 2 (Lehmer 1936, Erd®s 1956). For some constants ɛ1 1 and ɛ2, ɛ1log x2 √ ≤ P π2(x) ≤ x · exp{ɛ2 log x log log x}. In 1989, Pomerance =-=[6]-=- [11] proved tighter bounds for pseudoprime distribution. Theorem 3 (R.A. Mollin ed. 1989, Pomerance 1981). For the base-2 pseudoprime counting function, exp{(log x) 85 207 } ≤ P2(x) ≤ x · L(x) −1 log... |

8 |
On the number of Carmichael numbers up to x
- Harman
(Show Context)
Citation Context ...Pomerance a lower-bound for C(x) for x su ciently large [15]. Theorem 8 (Alford-Granville-Pomerance 1994). thus there are in nitely many Carmichael numbers. Recently, Harman improved this lower-bound =-=[18]-=-. Theorem 9 (Harman 2005). just below x 1 3 . It is not even known if C(x) > x 1 2 . C(x) > x 2 7 , (2.2.2) C(x) > x 0.332 , (2.2.3) 4Conjecture 10 (Pomerance 1981). The distribution of Carmichael nu... |

5 |
A New Lower Bound for the Pseudoprime Counting Function
- Pomerance
- 1982
(Show Context)
Citation Context ...d pseudoprimes with two distinct prime factors, and Pb,2(x) := #{n ≤ x : n = pq, p < q, Pb(n)}. Hence, lim x→∞ Pb,2(x) = 1 5 ɛ1 can be taken as 8 log 2 [10]. 2 The lower bound is accredited to Lehmer =-=[9]-=-. 1 Cx 2 log 2 , (2.1.3) x 3where C = 4T ∑ s≥1 ∑ r>s gcd(r,s)=1 T = 2 ∏ 1 − 2/d (1 − 1/d) d δ(rs)ρ(rs(r − s)) (rs) 3 2 ≈ 30.03, (2.1.4) 2 ≈ 1.32, (2.1.5) ρ(m) = ∏d − 1 , (2.1.6) d − 2 d|m ⎧ ⎪⎨ 2, if ... |

2 |
On the Converse of Fermat's Theorem
- Erd®s
- 1949
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Citation Context ... of mi if p divides mi but does not divide any mj for j < i. Lemma 13 (Erd®s 1949). Let n be a base-2 pseudoprime. For every k, there exist in nitely many squarefree base-2 pseudoprimes with ω(n) = k =-=[2]-=-. Lemma 14. We closely follow Erd®s' proof [2] of Lemma 13 to show that there exist innitely many squarefree base b pseudoprimes n for any b ≥ 2 with ω(n) = k distinct prime factors. Proof. Let {nj} ∞... |

2 | Tables of pseudoprimes and related data - Galway - 2002 |

2 | The Pseudoprimes below 2 64 - Galway - 2002 |

2 |
On Pseudoprimes which are Products of Distinct Primes
- Szymiczek
- 1967
(Show Context)
Citation Context .... Thus, pi·ni is squarefree and ω(pi·ni) = k. Moreover, every integer satisfying pi · ni is di erent because ni is composite, pi > ni, and pi ≡ 1 (mod (ni − 1)). Lemma 15. We extend Szymiczek's proof =-=[12]-=- to demonstrate that for any base b pseudoprime, b ≥ 2, having k ≥ 2 distinct prime factors and for x su ciently large, Pb,k+1(x) ≥ Pb,k(log b x). (3.1.1) Proof. Let n be a pseudoprime with k > 1 dist... |

2 |
G.H.Davies, The evaluation of κ3
- Chick
- 2008
(Show Context)
Citation Context ...− 3 ⎧ ⎪⎨ 2, if a ≡ b ≡ c ̸≡ 0 (mod 3); , (2.2.9) ⎪⎩ 1, if otherwise. and ωa,b,c(p) is the number of distinct residues modulo p represented by a, b, c. Recent provisional estimates by Chick and Davies =-=[14]-=- of the slowly converging in nite series κ3 suggest that κ3 = 27.05 which gives τ3 = 2087.5. In Table 4, it is evident that τ3 is in fact approaching 2087.5. We should note that we will use this value... |

1 |
There are In nitely Many Carmichael Numbers. Ann. of Math. 140
- Alford, Granville, et al.
- 1994
(Show Context)
Citation Context ...rd®s 1956). where k is a constant. { −k log x log log log x } C(x) < x · exp , (2.2.1) log log x In the other direction, Alford, Granville, and Pomerance a lower-bound for C(x) for x su ciently large =-=[15]-=-. Theorem 8 (Alford-Granville-Pomerance 1994). thus there are in nitely many Carmichael numbers. Recently, Harman improved this lower-bound [18]. Theorem 9 (Harman 2005). just below x 1 3 . It is not ... |

1 |
On pseudoprimes and Carmichael numbers
- Erd®s
- 1956
(Show Context)
Citation Context ...e 3 for it appears that C is slowly approaching its predicted constant value of 30.03 (2.1.4). 2.2 Carmichael Numbers 2.2.1 Carmichael Number Distribution Erd®s proved an upper-bound for C(x) in 1956 =-=[17]-=-. Theorem 7 (Erd®s 1956). where k is a constant. { −k log x log log log x } C(x) < x · exp , (2.2.1) log log x In the other direction, Alford, Granville, and Pomerance a lower-bound for C(x) for x su ... |

1 |
The Carmichael Numbers up to 10 to the 21. Eighth Algorithmic Number Theory Symposium ANTS-VIII May 17-22, 2008
- Pinch
(Show Context)
Citation Context ... let C(x) denote the number of Carmichael numbers ≤ x. Pomerance [8] conjectures that for all large x, {1+o(1)} log log log x 1− C(x) = x log log x . (1.0.3) Unfortunately, according to Richard Pinch =-=[19]-=-, Pomerance's conjecture does not appear to be well-supported by the data, an assertion which will be explained later in the discourse. As a result, we present an alternate conjecture based on a stron... |

1 |
Wagsta , jr. Pseudoprimes to 25
- Pomerance, Wagsta
- 1980
(Show Context)
Citation Context ...imes are rarer than the primes. Although this claim is rather elementary, Erd®s [11] went on to formulate an upper-bound for the pseudoprime counting function. The lower-bound is accredited to Lehmer =-=[10]-=-. Theorem 2 (Lehmer 1936, Erd®s 1956). For some constants ɛ1 and ɛ2, where ɛ1 √ can be taken as log x log log x}. 5 8 log 2 , ɛ1log x ≤ P π2(x) ≤ x · exp{ɛ2 In 1989, Pomerance [6] [11] proved tighter ... |

1 |
There are In nitely Many Carmichael Numbers
- Granville, Pomerance, et al.
- 1994
(Show Context)
Citation Context ...rmulate more e ective tests. Furthermore, there is little that is known about them; for instance, the in nitude of Carmichael numbers has only recently been proven by Alford, Granville, and Pomerance =-=[15]-=-. Let Pb(x) denote the number of base b pseudoprimes ≤ x; and let C(x) denote the number of Carmichael numbers ≤ x. In 1956, Erd®s [17] presented a heuristic argument that for su ciently large x, C(x)... |

1 | The Carmichael Numbers up to 10 21 . Proceedings Conference on Algorithmic Number Theory - Pinch - 2007 |