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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Two contradictory conjectures concerning Carmichael numbers
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by Andrew Granville and Carl Pomerance PDF
Math. Comp. 71 (2002), 883-908 Request permission

Abstract:

Erdős 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 Erdős 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 Erdős 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
  • MR Author ID: 76180
  • ORCID: 0000-0001-8088-1247
  • Email: andrew@math.uga.edu
  • Carl Pomerance
  • Affiliation: Fundamental Mathematics Research, Bell Laboratories, 600 Mountain Ave., Murray Hill, New Jersey 07974
  • MR Author ID: 140915
  • Email: carlp@research.bell-labs.com
  • 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.$^{1}$
  • © Copyright 2001 American Mathematical Society
  • Journal: Math. Comp. 71 (2002), 883-908
  • MSC (2000): Primary 11Y35, 11N60; Secondary 11N05, 11N37, 11N25, 11Y11
  • DOI: https://doi.org/10.1090/S0025-5718-01-01355-2
  • MathSciNet review: 1885636