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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 2024 MCQ for Mathematics of Computation is 1.78.

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A search for Fibonacci-Wieferich and Wolstenholme primes
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by Richard J. McIntosh and Eric L. Roettger;
Math. Comp. 76 (2007), 2087-2094
DOI: https://doi.org/10.1090/S0025-5718-07-01955-2
Published electronically: April 17, 2007

Abstract:

A prime $p$ is called a Fibonacci-Wieferich prime if $F_{p-({p\over 5})}\equiv 0\pmod {p^2}$, where $F_n$ is the $n$th Fibonacci number. We report that there exist no such primes $p<2\times 10^{14}$. A prime $p$ is called a Wolstenholme prime if ${2p-1\choose p-1}\equiv 1\pmod {p^4}$. To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes $p<10^9$. Wolstenholme, in 1862, proved that ${2p-1\choose p-1}\equiv 1\pmod {p^3}$ for all primes $p\ge 5$. It is estimated by a heuristic argument that the “probability” that $p$ is Fibonacci-Wieferich (independently: that $p$ is Wolstenholme) is about $1/p$. We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient $F_{p-({p\over 5})}/p$ modulo $p$.
References
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Bibliographic Information
  • Richard J. McIntosh
  • Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2
  • Email: mcintosh@math.uregina.ca
  • Eric L. Roettger
  • Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
  • Email: roettgee@math.ucalgary.ca
  • Received by editor(s): June 14, 2005
  • Received by editor(s) in revised form: May 19, 2006
  • Published electronically: April 17, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 76 (2007), 2087-2094
  • MSC (2000): Primary 11A07, 11A41, 11B39, 11Y99
  • DOI: https://doi.org/10.1090/S0025-5718-07-01955-2
  • MathSciNet review: 2336284