Some 3-adic congruences for binomial sums

Abstract

We prove some 3-adic congruences for binomial sums, which were conjectured by Zhi-Wei Sun. For example, for any integer m ≡ 1 (mod 3) and any positive integer n, we have

$\nu _3 \left( {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {\frac{1} {{m^k }}\left( \begin{gathered} 2k \hfill \\ k \hfill \\ \end{gathered} \right)} } \right) \geqslant \min \{ \nu _3 (n),\nu _3 (m - 1) - 1\} , $

where ν ₃(n) denotes the 3-adic order of n. In our proofs, we use several auxiliary combinatorial identities and a series converging to 0 over the 3-adic field.