FACTORS OF SUMS AND ALTERNATING SUMS INVOLVING BINOMIAL COEFFICIENTS AND POWERS OF INTEGERS

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers n(1),..., n(m), n(m)+1 = n(1), and any nonnegative integer r, there holds Sigma(n1)(k-0) epsilon(k)(2k + 1)(2r+1) Pi(m)(i=1) ((ni + ni+ 1+1)(ni - k)) equivalent to 0 (mod (n(1) + n(m) + 1) ((n1) (n1+nm))), and conjecture that for any nonnegative integer r and positive integer s such that r + s is odd, Sigma(n)(k=0) epsilon(k) (2k + 1)(r) (((2n)(n - k)) - ((2n)(n - k - 1)))(s) equivalent to 0 (mod ((2n)(n))), where epsilon = +/- 1.