A search for Wieferich and Wilson primes

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 $p$).