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### Abstract

Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding three-prime Carmichael numbers is described, with its implementation up to 10 24. Statistics relevant to the distribution of threeprime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [10]. 1.

### Citations

197 |
Probabilistic algorithm for testing primality
- Rabin
- 1980
(Show Context)
Citation Context ...s primitive iff H ≤ ABC; C ∗ 3(X) := #{n : n is a primitive C3N and n ≤ X}, and our data are consistent → 0 as X → ∞. C3(X) Let C := {n : n = pqr is a C3N and p ≡ q ≡ r ≡ −1 (mod 4)}; Rabin showed in =-=[15]-=- that the probability of any odd composite n passing the strong pseudoprime test for a randomly chosen base b is less than 1 4 , and that this bound is approached most closely when n ∈ C; and Pinch li... |

147 |
The book of prime number records
- Ribenboim
- 1988
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Citation Context ...the theory on Carmichael numbers and for algorithms to find them, including ours. For a background on Carmichael numbers and previous counts of Carmichaels up to increasing upper bounds see Ribenboim =-=[16]-=-, counts which have now culminated in Richard Pinch’s up to 1018 [13]. Our list up to 1024 for d = 3 may be found on the website of the Cambridge University Department of Pure Mathematics and Mathemat... |

21 |
Note on a new number theory function
- Carmichael
- 1910
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Citation Context ...hat αi = 1 for all i and lcm(p1 − 1, p2 − 1, . . .,pd − 1) divides (n − 1) is a necessary and sufficient condition for n to divide (an−a), but he did not exhibit any such number n. In 1910 Carmichael =-=[3]-=- showed that the above condition required d ≥ 3 and all pi to be odd, and gave four such numbers, the smallest of which was 561 = 3 ·11 ·17. In 1912 [4] he amplified his remarks and extended his list ... |

19 | The Carmichael numbers up to 1015
- Pinch
- 1993
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Citation Context ...lving simultaneously for q ′ and r ′ , and writing (2.8) ∆ := DE − P 2 , we get (2.9a) q ′ = P ′ (P + E) ∆ and . ∆ Beeger for d = 3 in 1950 [2] and Duparc for d ≥ 3 in 1952 [9] gave (2.9a), and Pinch =-=[12]-=- bases his first algorithm on (2.9). From (2.9), ∆ ≥ 1. Also Duparc showed (2.9b) r ′ = P ′ (P + D) Theorem 2.2. For any KN, 2 ≤ E ≤ P − 1. Proof. From the definition of a KN, E ≥ 2. Also r − q − 1 ≥ ... |

11 |
On Fermat T s Simple Theorem
- Chernick
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Citation Context ...here is some obvious reason why (almost) all members have at least one composite pi. 2.2. Chernick’s universal forms. The best known K-families are the “universal forms” described by Chernick in 1939 =-=[5]-=-, and it will be helpful to summarise his theory in our notation. Let n be any KN. Then we have (2.2) n ′ = d∏ (pi ′ +1)−1 = i=1 d∏ i=1 pi ′ + ∑ (p1 ′ p 2 ′ · · · p d−1 ′ )+. . .+ ∑ p1 ′ p 2 ′ + ∑ p1 ... |

10 | Density of Carmichael numbers with three prime factors
- Balasubramanian, Nagaraj
- 1997
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Citation Context ...e based on the loops of algorithm HII, ignoring the primality requirement on p, q, r. The best upper bound for C3(X) which has so far been proved is O(X 5 14 +o(1)), by Balasubramanian and Nagaraj in =-=[1]-=-. Table 2. Number and cumulative total of C3N’s with first prime p p 3 5 7 11 13 17 19 23 29 31 37 41 χ(p ) 1 3 6 0 5 2 2 1 2 7 5 7 T(p ) 1 4 10 10 15 17 19 20 22 29 34 41 p 43 47 53 59 61 67 71 73 79... |

7 | Probléme Chinois. L’intermédiaire des mathématiciens - Korselt |

6 |
On composite numbers n for which a n−1 ≡ 1 mod n for every a prime to n, Scripta Mathematica 16
- Beeger
- 1950
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Citation Context ....7) Dq ′ − Pr ′ = P ′ , and similarly Er ′ − Pq ′ = P ′ . Solving simultaneously for q ′ and r ′ , and writing (2.8) ∆ := DE − P 2 , we get (2.9a) q ′ = P ′ (P + E) ∆ and . ∆ Beeger for d = 3 in 1950 =-=[2]-=- and Duparc for d ≥ 3 in 1952 [9] gave (2.9a), and Pinch [12] bases his first algorithm on (2.9). From (2.9), ∆ ≥ 1. Also Duparc showed (2.9b) r ′ = P ′ (P + D) Theorem 2.2. For any KN, 2 ≤ E ≤ P − 1.... |

6 |
On Carmichael numbers, Simon Stevin 29
- Duparc
- 1952
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Citation Context ...arly Er ′ − Pq ′ = P ′ . Solving simultaneously for q ′ and r ′ , and writing (2.8) ∆ := DE − P 2 , we get (2.9a) q ′ = P ′ (P + E) ∆ and . ∆ Beeger for d = 3 in 1950 [2] and Duparc for d ≥ 3 in 1952 =-=[9]-=- gave (2.9a), and Pinch [12] bases his first algorithm on (2.9). From (2.9), ∆ ≥ 1. Also Duparc showed (2.9b) r ′ = P ′ (P + D) Theorem 2.2. For any KN, 2 ≤ E ≤ P − 1. Proof. From the definition of a ... |

4 |
composite numbers P which satisfy the Fermat congruence aP −1
- On
- 1912
(Show Context)
Citation Context ...t exhibit any such number n. In 1910 Carmichael [3] showed that the above condition required d ≥ 3 and all pi to be odd, and gave four such numbers, the smallest of which was 561 = 3 ·11 ·17. In 1912 =-=[4]-=- he amplified his remarks and extended his list to fifteen such numbers, including one with d = 4 (although very curiously he reconsidered and rejected 561!) Korselt’s criterion, stated above, is the ... |

3 | Numbers of the form (6m+1)(12m+1)(18m+1
- Dubner
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Citation Context ...took PII about 74 hours and HII about 31 2 minutes. We are grateful to Harvey Dubner for collaboration which gave a further partial check. Let C † 3 (X) := #{n : n is a C3N with A = 1, and n ≤ X}. In =-=[8]-=- Dubner 3 (10N) finds C † 3 (10N) up to N = 20, and suggests that for a “wide range of N”, C† C3(10N ) ≏ 0.644. He uses an entirely different algorithm for C † 3 (X), based on relevant (1, B, C) value... |

2 |
G.H.Davies, The evaluation of κ3
- Chick
- 2008
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Citation Context ... program gave it. He also kindly put me in touch with Carl Pomerance, who sent me the first preprint of [10] and invited Gordon Davies and me to attempt the awkward evaluation of the constant κ3 (see =-=[6]-=-). Some months later when Carl asked us for any counts we had beyond X = 10 18 , Matthew had got to X = 10 20 , but had not yet done any checks; it later emerged that a problem in the program was by X... |

2 |
Table of Carmichael numbers to 10 9
- Swift
- 1975
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Citation Context ... than Theorem 3.2. They were chiefly concerned to show that the number of CN’s for given P is finite. Swift stated the first result of Theorem 3.3 for d = 3 in 1975, but his proof is not published in =-=[17]-=-. For a K3-family attaining these upper bounds for r and n given P, we simply put P = 2t + 1 in n = N3(P) = P · Q3(P) · R3(P). We note that Q2(3) = Q3(3) = 11 and R2(3) = R3(3) = 17, so the smallest C... |