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Mathematics of Computation

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Two contradictory conjectures concerning Carmichael numbers

Authors: Andrew Granville and Carl Pomerance
Journal: Math. Comp. 71 (2002), 883-908
MSC (2000): Primary 11Y35, 11N60; Secondary 11N05, 11N37, 11N25, 11Y11
Published electronically: October 4, 2001
MathSciNet review: 1885636
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Abstract: Erdos conjectured that there are $x^{1-o(1)}$ 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 $x^{2/7}$ Carmichael numbers up to $x$, and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdos 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 Erdos and the results of Alford, Granville and Pomerance.

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Additional Information

Andrew Granville
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Carl Pomerance
Affiliation: Fundamental Mathematics Research, Bell Laboratories, 600 Mountain Ave., Murray Hill, New Jersey 07974

Received by editor(s): November 11, 1999
Received by editor(s) in revised form: July 25, 2000
Published electronically: October 4, 2001
Additional Notes: The first author is a Presidential Faculty Fellow. Both authors were supported, in part, by the National Science Foundation
Dedicated: Dedicated to the two conjecturers, Paul Erdős and Dan Shanks. We miss them both.$^{1}$
Article copyright: © Copyright 2001 American Mathematical Society