Wieferich primes (statement)

Definition. $p$ is Wieferich if $2^{p-1}\equiv1\pmod{p^2}$, i.e. $p^2\mid2^{p-1}-1$.

1093. It is known (and can be verified by fast exponentiation) that $2^{1092}\equiv1\pmod{1093^2}$, so $1093$ is a Wieferich prime.

Conclusion. Only two Wieferich primes are known: $1093$ and $3511$.