I'm interested in the multiplicative order 2 mod $p$ where $p$ is a Wieferich prime. The only two known are 1093 and 3511, which have orders 364 and 1755 respectively. It is known that there are no Wieferich primes less than $\ell = 5171487\cdot10^{11}$ (leaving this as a variable since the search bounds will increase over time).
By definition of Wieferich primes the order of 2 divides $p-1$ which is also an upper bound. For $2^m\equiv1\pmod{p}$ we must have $2^m \ge p+1$ and so $\log_2(p+1)$ is a lower bound. But $\lceil\log_2{\ell+1}\rceil=59$ which is an unsatisfactory lower bound for my purpose. ($10^7$ would be great.) Can it be proved that there is no Wieferich prime $p>3511$ such that the order of 2 mod $p$ is less than $x$ for some decent-sized $x$?