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 is an element of Z with 2h - 1 = 0(modp). For a is an element of Z(+) and p(a) > 3, we show that Sigma(pa-1)(k=0) (hpa - 1 k) (2k k)(- (h/2))k = 0(modp(a+1)). Also, for any n is an element of Z(+), we have nu(p) (hn - 1 k) (2k k)(- (h/2))(k)) >= nu(p) (n), where nu(p) (n) denotes the p-adic order of n. For any integer m = 0(modp) and positive integer n, we have (1/ pn) (Sigma(pn-1)(k=0) (pn - 1 k) ((2k k)/(- m)(k) (m(m - 4)/p)Sigma(n- 1)(k=0) (n - 1 k) (2k k) /(- m)k) is an element of Z(p), where (-) is the Legendre symbol and Z(p) is the ring of p-adic integers.