I need to prove that:
If $p$ is prime greater than or equal to five, then $pB_{p-1}$ belongs to the p-integers and more over:
$$pB_{p-1} \equiv -1 \pmod p$$ Hint:Put $N=p$ in the Faulhaber´s formula: $S_{j}(N-1)= \displaystyle\sum_{k=o}^j \frac{1}{(j-k+1)}{j \choose k}N^{j-k+1}B_{k}$ and reduce everything mod p and use the Fermat´s little theorem.
But I do not know how to reduce this: $S_{j}(p-1)= \displaystyle\sum_{k=o}^j \frac{1}{(j-k+1)}{j \choose k}p^{j-k+1}B_{k}$ to mod p, and I have problems trying to prove that $pB_{p-1}$ belongs to the p-integers, Can you help to prove the theorem please, Thank you.