The p-Adic Valuations of Sums of Binomial Coefficients

In this paper, we prove three supercongruences on sums of binomial coefficients conjectured by Z.-W. Sun. Let p be an odd prime and let h∈ℤ with 2h−1≡0modp. For a∈ℤ+ and pa>3, we show that ∑k=0pa−1hpa−1k2kk−h/2k≡0modpa+1. Also, for any n∈ℤ+, we have νp∑k=0n−1hn−1k2kk−h/2k≥νpn, where νpn denotes the p-adic order of n. For any integer m≡0modp and positive integer n, we have 1/pn∑k=0pn−1pn−1k2kk/−mk−mm−4/p∑k=0n−1n−1k2kk/−mk∈ℤp, where −˙ is the Legendre symbol and ℤp is the ring of p-adic integers.