## Two contradictory conjectures concerning Carmichael numbers

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@MISC{Granville_twocontradictory,

author = {Andrew Granville and Carl Pomerance},

title = {Two contradictory conjectures concerning Carmichael numbers},

year = {}

}

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### Abstract

Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].

### Citations

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Some Problems of ’Partitio Numerorum.’ III. On the Expression of a Number as a Sum of Primes
- Hardy, Littlewood
- 1922
(Show Context)
Citation Context ...ppropriately applied to primitive Carmichael numbers, thus partially resolving their contradictory conjectures in a way that makes both of them right. We begin by examining the data made available in =-=[12]-=-, [13], making several easy observations and recalling some known facts. By computing thesrst few examples one quickly observes that Carmichael numbers seem to all have at least three prime factors. T... |

79 |
On a problem of Oppenheim concerning “factorisatio numerorum
- Canfield, Erdös, et al.
- 1983
(Show Context)
Citation Context ...t the number of members of S is N= log(x1=k), where N is the number of integers x1=k which divide L. Thus, we conjecture that #S x1=k=u(1+o(1))u, where u = log(x1=k)= log(log x= log log x), see =-=[5]-=-. A calculation like the one above gives Conjecture 2a. With more care one can optimize the above argument, and conjecture [17], [19] C(x) = x1f1+o(1)g log log log x= log log x; one can also prove th... |

61 |
certaines hypothèses concernant les nombres premiers, Acta Arithmetica 4(1958), 185 – 208. 400 E.Remington Dr
- Schinzel, Sierpiński, et al.
(Show Context)
Citation Context ... Sylvester and Dickson, and although little is yet proved we do now have a good conjectural understanding of how often these are prime, thanks to Hardy and Littlewood [12] and Schinzel and Sierpinski =-=[22]-=-, [23]: Prime Triplets Conjecture. Let a1t+ b1; a2t+ b2 and a3t+ b3 be distinct linear polynomials, with integer coecients, where each ai is positive and coprime to bi. If there is an integer r such ... |

35 | The Gaussian law of errors in the theory of additive number-theoretic functions - Erdős, Kac - 1940 |

33 |
Average Case Error Estimates for the Strong Probable Prime Test
- Damgård, Landrock, et al.
- 1993
(Show Context)
Citation Context ...ture, based on the available data, and because he pointed out that it would be far easier to analyze the reliability of pseudoprime tests if there were very few pseudoprimes (however, the analysis in =-=[6]-=- is suitable for Shanks's requirements) { see Section 4 for details of Shanks's remarks. So what explains this discrepancy between the computational evidence and the predicted asymptotic behavior, for... |

23 |
On the Distribution of Pseudoprimes
- Pomerance
- 1981
(Show Context)
Citation Context ... This leads to the following single formula which implies Conjectures 1 and 2: Conjecture 3. If k is an integer in the range 3 k y := log x= log log x, then Ck(x) = x1=k k! ky (log log x)O(y): In =-=[17]-=- a heuristic argument is given that C(x) = x1(x); where (x) = (1+o(1)) log log log x= log log x. Further, it is seen from the argument in [17] that Ck(x) is of similar magnitude when k log x=(log... |

20 | The Carmichael numbers up to 1015
- Pinch
- 1993
(Show Context)
Citation Context ...at very few of them are imprimitive. This supports our conjecture that C0(x) = o(C(x)). Moreover Theorems 3 and 5b together suggest that C0k(x) = o(Ck(x)) if k log log x. Using Richard Pinch's data =-=[14]-=-, [15], [16] and unpublished calculations of Chick and Davies and of Williams, we can see how these last two conjectures compare with the known Carmichael numbers. For x = 106; 107; : : : ; 1016 we wr... |

16 |
The pseudoprimes to 25
- Pomerance, Selfridge, et al.
- 1980
(Show Context)
Citation Context ...of Section 2, we understand why one might be led to Conjecture 1, in the case k = 3. There have been several results which imply good upper bounds for C3(x). In 1980, Pomerance, Selfridge and Wagsta =-=[21]-=- showed that C3(x) x2=3. In 1993 Damgard, Landrock and Pomerance [6] gave an explicit estimate of the shape C3(x) x1=2+o(1). In 1995, S.W. Graham (unpublished) showed that C3(x) x2=5+o(1), and ... |

13 | There are infinitely many Carmichael - Alford, Granville, et al. - 1994 |

11 |
Two methods in elementary analytic number theory
- Pomerance
(Show Context)
Citation Context ... x1=k=u(1+o(1))u, where u = log(x1=k)= log(log x= log log x), see [5]. A calculation like the one above gives Conjecture 2a. With more care one can optimize the above argument, and conjecture [17], =-=[19]-=- C(x) = x1f1+o(1)g log log log x= log log x; one can also prove that the implicit upper bound here holds (see [17], [19]). We expect that the same estimate holds for 2(x), the number of base 2 pseud... |

10 | Density of Carmichael numbers with three prime factors
- Balasubramanian, Nagaraj
- 1997
(Show Context)
Citation Context ... trying to explain the reasoning that led to this conjecture, especially since there have now been several papers with partial results towards Conjecture 1; most recently, Balasubramanian and Nagaraj =-=[4]-=- have shown that C3(x) x5=14+o(1).) 886 ANDREW GRANVILLE AND CARL POMERANCE In Theorem 7 we prove Ck(x) x2=3+ok(1), though we would like to improve this to Ck(x) x1=2+ok(1), for eachsxed k. Note... |

7 |
Local densities over integers free of large prime factors
- Ivic, Tenenbaum
- 1986
(Show Context)
Citation Context ...iately applied to primitive Carmichael numbers, thus partially resolving their contradictory conjectures in a way that makes both of them right. We begin by examining the data made available in [12], =-=[13]-=-, making several easy observations and recalling some known facts. By computing thesrst few examples one quickly observes that Carmichael numbers seem to all have at least three prime factors. This is... |

6 |
Primality Testing and Carmichael Numbers
- Granville
- 1992
(Show Context)
Citation Context ...s set of primes, wesnd that 41041 = 7 11 13 41; CARMICHAEL NUMBERS 891 172081 = 7 13 31 61; 852841 = 11 31 41 61 are all 1 (mod 120) and so are all Carmichael numbers. Alford (see =-=[11]-=-) took a large value for L, determined many primes p for which p 1 divides L, and then established that there are at least 2128 1 Carmichael numbers made up from them | this was the inspiration fo... |

5 | On the difficulty of finding reliable witnesses - Alford, Granville, et al. - 1994 |

3 |
On pseudoprimes and Carmichael numbers, Publ
- Erdos
- 1956
(Show Context)
Citation Context ...believed) assumptions about the distribution of primes in arithmetic progressions, it is shown in Theorem 4 of [2] that there are x1o(1) Carmichael numbers up to x, as had been conjectured by Erd}os =-=[8]-=-. However, for x = 10n for n up to 16 (which is as far as has been computed [15]), there are less than x0:337 Carmichael numbers up to x and, extrapolating Received by the editor November 11, 1999 and... |

3 |
The Carmichael numbers up
- Pinch
(Show Context)
Citation Context ...ns, it is shown in Theorem 4 of [2] that there are x1o(1) Carmichael numbers up to x, as had been conjectured by Erd}os [8]. However, for x = 10n for n up to 16 (which is as far as has been computed =-=[15]-=-), there are less than x0:337 Carmichael numbers up to x and, extrapolating Received by the editor November 11, 1999 and, in revised form, July 25, 2000. 2000 Mathematics Subject Classication. Primar... |

3 |
There are in many Carmichael
- Alford, Granville, et al.
- 1994
(Show Context)
Citation Context ...found by Carmichael in 1910. It was recently shown that there are innitely many Carmichael numbers; in fact, that there are more than x2=7 Carmichael numbers up to x, once x is suciently large (see =-=[2]-=-). Moreover, under certain (widely-believed) assumptions about the distribution of primes in arithmetic progressions, it is shown in Theorem 4 of [2] that there are x1o(1) Carmichael numbers up to x,... |

2 |
The density of pseudoprimes with two prime factors
- Galway
(Show Context)
Citation Context ...uasive that the numberssandsare tending to a common limit which is about 2100, they at least suggest that our heuristic is not too wildly wrong. We can compare our conjecture with that made by Galway =-=[10]-=- for the number of 2-pseudoprimes x with exactly two prime factors. Note that n = (ag+ 1)(bg+ 1) is a 2-pseudoprime, where each of the two factors are primes, and (a; b) = 1, if and only if 2g 1 (... |

2 |
certaines hypotheses concernant les nombres premiers
- Sur
- 1959
(Show Context)
Citation Context ...ster and Dickson, and although little is yet proved we do now have a good conjectural understanding of how often these are prime, thanks to Hardy and Littlewood [12] and Schinzel and Sierpinski [22], =-=[23]-=-: Prime Triplets Conjecture. Let a1t+ b1; a2t+ b2 and a3t+ b3 be distinct linear polynomials, with integer coecients, where each ai is positive and coprime to bi. If there is an integer r such that (... |

2 |
Solved and unsolved problems in number theory, 3rd ed
- Shanks
- 1985
(Show Context)
Citation Context ...e data to hand, it seems unlikely that there will be more than x1=2 Carmichael numbers up to x for any x < 10100. In this article we are interested in this strange phenomenom,srst discussed by Shanks =-=[24]-=-. He showed skepticism of Erd}os's conjecture, based on the available data, and because he pointed out that it would be far easier to analyze the reliability of pseudoprime tests if there were very fe... |

2 | On the diculty of reliable witnesses - Alford, Granville, et al. - 1995 |

1 |
Carmichael numbers with exactly k prime factors
- Alford, Grantham
(Show Context)
Citation Context ... The consequence in Corollary 1 rests on there being at least one Carmichael number with exactly k prime factors, which is by no means guaranteed, a priori (though it is known for 3 k 1; 000; 000 =-=[1]-=-). We now construct a family with exactly k prime factors (assuming the Hardy-Littlewood Conjecture holds), for each k 3, by modifying an idea of Euclid: Choose n 2 so that k = 2n 1 or 2n. Let a... |

1 |
The pseudoprimes up
- Pinch
(Show Context)
Citation Context ...of them are imprimitive. This supports our conjecture that C0(x) = o(C(x)). Moreover Theorems 3 and 5b together suggest that C0k(x) = o(Ck(x)) if k log log x. Using Richard Pinch's data [14], [15], =-=[16]-=- and unpublished calculations of Chick and Davies and of Williams, we can see how these last two conjectures compare with the known Carmichael numbers. For x = 106; 107; : : : ; 1016 we write the numb... |

1 | The pseudoprimes to 25·109 ,Math.Comp - Pomerance, Selfridge, et al. - 1980 |