Proof of some congruences conjectured by Z.-W. Sun

In this paper, we show that for any prime Formula: see text, Formula: see text and Formula: see text where Formula: see text denotes the Bernoulli polynomial of degree Formula: see text. And we prove that Formula: see text and Formula: see text where Formula: see text stands for the Formula: see textth Bernoulli number. This confirms several conjectures of Z.-W. Sun.