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- A. Granville and C. Pomerance, "Two contradictory conjectures concerning Carmichael numbers," Math. Comp., 71 (2002) 883--908. MR 1 885 636
Erdös conjectured that there are x1-o(1) Carmichael numbers up to x, whereas Shanks was skeptical as to whether one might even find an x up to which there are more than sqrt(x) Carmichael numbers. Alford, Granville and Pomerance showed that there are more than x2/7 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), and so we herein derive conjectures that are consistent with Shanks's observations, while fitting in with the viewpoint of Erdös and the results of Alford, Granville and Pomerance.