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A search for Wieferich and Wilson primes
Authors:Richard Crandall  Karl Dilcher  Carl Pomerance
Institution:Center for Advanced Computation, Reed College, Portland, Oregon 97202 ; Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada ; Department of Mathematics, University of Georgia, Athens, Georgia 30602
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 $p$).

Keywords:Wieferich primes  Wilson primes  Fermat quotients  Wilson quotients  factorial evaluation
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