| Science Foundation. (2008) | |||||||||||||||
Abstract | |||||||||||||||
| Abstract. Erd}os [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 Erd}os 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 ofErd}os [8] and the results of [2,3]. 1 Dan Shanks passed away onSeptember 6th, 1996, and Paul Erd}os two weeks later. The rst | |||||||||||||||
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