## RICHARD G.E. PINCH (1998)

### BibTeX

@MISC{98richardg.e.,

author = {},

title = {RICHARD G.E. PINCH},

year = {1998}

}

### OpenURL

### Abstract

Abstract. We extend our previous computations to show that there are 246683 Carmichael numbers up to 10 16. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael numbers. 1.

### Citations

20 |
On the distribution of pseudoprimes
- Pomerance
- 1981
(Show Context)
Citation Context ...) ) logX log log log X . log log X They proved that liminf k ≥ 1 and suggested that limsupk might be 2, although they also observed that within the range of their tables k(X) is decreasing: Pomerance =-=[3]-=-,[4] gave a heuristic argument suggesting that limk = 1. The decrease in k is reversed between 10 13 and 10 14 : see Figure 1. We find no clear support from our computations for any conjecture on a li... |

18 | The Carmichael numbers up to 10 15
- Pinch
- 1993
(Show Context)
Citation Context ... p − 1|N − 1 for every prime p dividing N: conversely, any such N must be a Carmichael number. For background on Carmichael numbers and details of previous computations we refer to our previous paper =-=[2]-=-: in that paper we described the computation of the Carmichael numbers up to 10 15 and presented some statistics. These computations have since been extended to 10 16 , using the same techniques, and ... |

8 |
jr, The pseudoprimes up to 25.10
- Pomerance, Selfridge, et al.
- 1980
(Show Context)
Citation Context ... smallest Carmichael numbers with d prime factors for d up to 20. The results are given in Table 3. In Table 4 and Figure 1 we tabulate the function k(X), defined by Pomerance, Selfridge and Wagstaff =-=[5]-=- by C(X) = X exp ( −k(X) ) logX log log log X . log log X They proved that liminf k ≥ 1 and suggested that limsupk might be 2, although they also observed that within the range of their tables k(X) is... |

5 |
Review 13[9] — table of Carmichael numbers to 10 9
- Swift
- 1975
(Show Context)
Citation Context ... 55012 86696 60150 16348 1436 23 246683 Table 2. Values of C(X) and C(d, X) for d ≤ 10 and X in powers of 10 up to 1016 . In Table 4 we also give the ratios C(10 n )/C ( 10 n−1) investigated by Swift =-=[6]-=-. Swift’s ratio, again initially decreasing, also increases again before 10 15 . In Table 5 and Figure 2 we see that within the range of our computations C(X) is a slowly growing power of X: about X 0... |