Results 1 
2 of
2
Two contradictory conjectures concerning Carmichael numbers
"... 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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
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 persontoperson 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].
Carmichael Numbers of the form (6m + 1)(12m + 1)(18m + 1)
, 2002
"... Numbers of the form (6m + 1)(12m + 1)(18m + 1) where all three factors are simultaneously prime are the best known examples of Carmichael numbers. In this paper we tabulate the counts of such numbers up to 10 for each n 42. We also derive a function for estimating these counts that is remarkably ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Numbers of the form (6m + 1)(12m + 1)(18m + 1) where all three factors are simultaneously prime are the best known examples of Carmichael numbers. In this paper we tabulate the counts of such numbers up to 10 for each n 42. We also derive a function for estimating these counts that is remarkably accurate.