- $$ pB_{p-1}+(p-1)!(p-1) \equiv 0\mod p^2.$$
- $$ pB_{p-1}-p-(p-1)! \equiv 0\mod p^2.$$
I have (I believe) a proof for the above supercongruences, where $p$ is an odd prime, and $B_{p-1}$ is a Bernoulli number. I guess it is already known, but I could not find a source, I have checked on Wikipedia only though.
The demonstration that I have found uses the follwing relationship, involving Binomial coefficients, Bernoulli and Stirling numbers of both kinds. $$ \binom{n+1}{j}B_j=(-1)^{n+1}(n+1)\sum_{1\le k\le n+1}\frac{(-1)^{k}}{k} \begin{Bmatrix}n+1\\k\end{Bmatrix}\left[\begin{array}{ccc}k \\ n+1-j \end{array}\right]$$ This is quite complicated and I would be interested in a more straigthforward proof.
EDIT
For $p=17,1733, 18433,?.. $,the congruence (1) holds modulo $p^3$. I could not find that sequence in oeis.org, and I just added it.
For $p=103,839,2237,?.. $, the congruence (2) holds modulo $p^3$. The sequence apparently is that of the Lerch primes, but for the first one ($p=3$). Why is that?
These supercongruences sound like Wolstenholme's.
I also believe that each of the above supercongruence is a sufficient condition for $p$ to be odd prime. Please check.
I am answering my own question: I just found this paper, where another proof of (2) is given, along with the connection with Lerch primes when (2) holds modulo $p^3$