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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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A search for Wieferich and Wilson primes
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by Richard Crandall, Karl Dilcher and Carl Pomerance PDF
Math. Comp. 66 (1997), 433-449 Request permission

Abstract:

An odd prime $p$ is called a Wieferich prime if \begin{equation*}2^{p-1} \equiv 1 \pmod {p^{2}};\end{equation*} alternatively, a Wilson prime if \begin{equation*}(p-1)! \equiv -1 \pmod { p^{2}}.\end{equation*} To date, the only known Wieferich primes are $p = 1093$ and $3511$, while the only known Wilson primes are $p = 5, 13$, and $563$. We report that there exist no new Wieferich primes $p < 4 \times 10^{12}$, and no new Wilson primes $p < 5 \times 10^{8}$. It is elementary that both defining congruences above hold merely (mod $p$), and it is sometimes estimated on heuristic grounds that the “probability" that $p$ is Wieferich (independently: that $p$ is Wilson) is about $1/p$. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod $p$).
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Additional Information
  • Richard Crandall
  • Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon 97202
  • Email: crandall@reed.edu
  • Karl Dilcher
  • Affiliation: Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada
  • Email: dilcher@cs.dal.ca
  • Carl Pomerance
  • Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
  • MR Author ID: 140915
  • Email: carl@ada.math.uga.edu
  • Received by editor(s): May 19, 1995
  • Received by editor(s) in revised form: November 27, 1995, and January 26, 1996
  • Additional Notes: The second author was supported in part by a grant from NSERC. The third author was supported in part by an NSF grant.
  • © Copyright 1997 American Mathematical Society
  • Journal: Math. Comp. 66 (1997), 433-449
  • MSC (1991): Primary 11A07; Secondary 11Y35, 11--04
  • DOI: https://doi.org/10.1090/S0025-5718-97-00791-6
  • MathSciNet review: 1372002