Abstract:

We prove several congruences modulo a power of a prime, such as

$$\sum_{0<k_1<\cdots<k_{n}<p}\genfrac{(}{)}{}{}{p-k_{n}}{3} \f... ...eft({1/3}\right) \pmod{p}, &\text{if $n$\ is even,} \end{cases}$$

where $n$ is a positive integer and $p$ is a prime such that $p>\max(n+1,3)$.