Some 3-adic congruences for binomial sums
Some 3-adic congruences for binomial sums
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DOI：
10.1007/s11425-013-4723-9
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发表时间：
2012-03
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通讯作者：
Yong Zhang;H. Pan
中科院分区：
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作者：
Yong Zhang;H. Pan
We prove some 3-adic congruences for binomial sums, which were conjectured by Zhi-Wei Sun. For example, for any integerm≡ 1 (mod 3) and any positive integern, 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\} , $ whereν3(n) denotes the 3-adic order ofn. In our proofs, we use several auxiliary combinatorial identities and a series converging to 0 over the 3-adic field.
We prove some 3-adic congruences for binomial sums, which were conjectured by Zhi-Wei Sun. For example, for any integerm≡ 1 (mod 3) and any positive integern, 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\} , $ whereν3(n) denotes the 3-adic order ofn. In our proofs, we use several auxiliary combinatorial identities and a series converging to 0 over the 3-adic field.