A NOTE ON THE CONGRUENCES WITH SUMS OF POWERS OF BINOMIAL COEFFICIENTS

Let p >= 7 be a prime, l >= 0 be an integer and k, m be two positive integers, we obtain the following congruences, Sigma((l+1)p-1)(s=lp) ((kp-1)(s))(m) {((k-1)(l))(m) 2(km(p-1)) (mod p(3)), if 2 inverted iota m, ((k-1)(l))(m) ((kmp-2)(p-1)) (mod p(4)), if 2 vertical bar m; and Sigma((l+1)p-1)(s=lp) (-1)(s)((kp-1)(s))(m) {(-1)(l)((k-1)(l))(m) 2(km(p-1)) (mod p(3)), if 2 inverted iota m, (-1)(l)((k-1)(l))(m) ((kmp-2)(p-1)) (mod p(4)), if 2 vertical bar m. Let p and q are distinct odd primes and k be a positive integer, we have ((kpq-1)((pq-1)/2)) ((kp-1)((p-1)/2)) ((kq-1)((q-1)/2)) (mod pq).