Some 3-adic congruences for binomial sums |
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Authors: | Yong Zhang Hao Pan |
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Institution: | 1. Department of Basic Course, Nanjing Institute of Technology, Nanjing, 211167, China 2. Department of Mathematics, Nanjing University, Nanjing, 210093, China
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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. |
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Keywords: | -adic order binomial sum |
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