On some conjectural congruences

Abstract

In this paper, we confirm some congruences conjectured by V.J.W. Guo and M.J. Schlosser recently. For example, we show that for primes $p>3$, $$\sum _{k=0}^{p-1}(2pk-2k-1)\frac{{\left(\frac{-1}{p-1}\right)}_k^{2p-2}}{(k!{)}^{2p-2}}\equiv 0(\mathrm{mod}{p}^5).$$

8 pages