## 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