What is the expected magnitude of the third Wieferich-prime? A Wieferich-prime is a prime $p$ with the property $$2^{p-1}\equiv 1\ \pmod{p^2} $$
The only known Wieferich-primes are $1093$ and $3511$. Various search limits
are mentioned in the internet. Here :
https://en.wikipedia.org/wiki/Wieferich_prime
the largest search limit is claimed to be $6.7\times 10^{15}$, but $3\times 10^{17}$ is also mentioned.

Is the later limit already reached or in progress ?

Despite the rarity of Wieferich-primes it is conjectured that there are
infinite many.

What is the expected magnitude of the third Wieferich-prime considering the
    largest search limits ?

 A: I find the literal answer to the second question surprising and funny.
One might expect that the residue class $\frac{2^{p-1}-1}{p}\mod p$ should act as if it is determined uniformly at random. So a random prime $p$ is Wieferich with probability $\frac{1}{p}$. Since the probability that a random integer $n$ is prime is about $\frac{1}{\log n}$, this suggests that the number of Wieferich primes less than $x$ should grow like
$$
\sum_{n\leq x}\frac{1}{n\log n}\sim \log\log x.
$$
In particular, there are expected to be infinitely many Wieferich primes because $\log\log x\to\infty$ as $x\to\infty$.
If we pick some integer $N$, how large is the next largest Wieferich prime after $N$? The "probability" that the next largest Wieferich prime is $n$ is about
$$
\frac{1}{n\log n} \cdot \prod_{m=N+1}^{n-1}\left(1-\frac{1}{m\log m}\right)\sim\frac{1}{n\log n}\frac{\log N}{\log n}.
$$
The expected value of the magnitude of the next Wieferich prime after $N$ is
$$
\sum_{n=N+1}^\infty n\cdot\left(\frac{1}{n\log n}\frac{\log N}{\log n}\right)=\log N\sum_{n=N+1}^\infty\frac{1}{(\log n)^2}=+\infty.
$$
So even though we expect there are infinitely many Wieferich primes, the expected magnitude of the next one is infinite!
