On 2-adic orders of some binomial sums

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$\sum_{k=-n}^n\binom{2n}{n-k}k^{2r}$$ is divisible by $2^{2n-\min\{\alpha(n),\alpha(r)\}}$, where $\alpha(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.Comment: 6 page

A note on the Polignac numbers

Suppose that $k\geq 3.5\times 10^6$ and \hH=\{h_1,\ldots,h_{k_0}\} is admissible. Then for any $m\geq 1$, the set $$\{m(h_j-h_i):\,h_i<h_j\}$$ contains at least one Polignac number