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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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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


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:
  • Karl Dilcher
  • Affiliation: Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada
  • Email:
  • Carl Pomerance
  • Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
  • MR Author ID: 140915
  • Email:
  • 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:
  • MathSciNet review: 1372002