Proof of a conjecture of Sun and its extension by Guo.

In this paper, we mainly prove the following result: For any positive integers l and n and nonnegative integer k with k ≤ n - 1 , we have (2 l - 1) ! ! ∑ m = k n - 1 (2 m + 1) 2 l - 1 m + k 2 k 2 k k 2 ≡ 0 mod n 2 n - 1 k n + k k . This confirms a conjecture of Sun and its extension by Guo. [ABSTRACT FROM AUTHOR]