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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Constructing Carmichael numbers through improved subset-product algorithms
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by \fbox{W. R. } Alford, Jon Grantham, Steven Hayman and Andrew Shallue PDF
Math. Comp. 83 (2014), 899-915 Request permission


We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with $k$ prime factors for every $k$ between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes $p$ with the property that $p-1$ divides a highly composite $\Lambda$.
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Additional Information
  • Jon Grantham
  • Affiliation: Institute for Defense Analyses, Center for Computing Sciences, 17100 Science Drive, Bowie, Maryland 20715
  • Email:
  • Steven Hayman
  • Affiliation: 1331 East Washington Street, Unit A, Greenville, South Carolina 29607
  • Email:
  • Andrew Shallue
  • Affiliation: Illinois Wesleyan University, 1312 Park St., Bloomington, Illinois 61701
  • MR Author ID: 805175
  • Email:
  • Received by editor(s): December 15, 2011
  • Received by editor(s) in revised form: May 24, 2012
  • Published electronically: July 9, 2013
  • Additional Notes: W. R. Alford passed away in 2003
    This research was supported by an Illinois Wesleyan University grant
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 899-915
  • MSC (2010): Primary 11Y16
  • DOI:
  • MathSciNet review: 3143697