| 二项式系数的更多整除性质 |
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| 引用本文： | 肖嘉琪,贺玉清,袁平之.二项式系数的更多整除性质[J].数学研究及应用,2022,42(6):561-579. |
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| 作者姓名： | 肖嘉琪 贺玉清 袁平之 |
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| 作者单位： | 华南师范大学数学科学学院, 广东 广州 510631 |
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| 基金项目： | 国家自然科学基金(Grant No.12171163). |
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| 摘 要： | 设$a$, $b$和$n$为正整数,且$a>b$,我们证明了下面的整除性质: 对所有正整数$n$, 我们有$$(2bn+1)(2bn+3)(2bn+5){2bn\choose bn}\Big|15(a-b)(3a-b)(5a-b)(5a-3b){2an \choose an}{an\choose bn},$$ 上述整除式推广了杨全会一文中的相关结论.且对所有正整数$n$,我们证明了下面的整除性质:$$(6n+1){4n\choose n}\Big|{12n\choose 6n}{2n\choose n},\ (12n+1){5n\choose n}\Big|{15n\choose 3n}{3n-1\choose n-1},$$ $$(18n+1){12n\choose 9n}{8n\choose 2n}\Big| {24n\choose 18n}{4n\choose 2n}{6n\choose 3n}.$$更多类似的整除性质可以给出.
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| 关 键 词： | 二项式系数 p-进指数 整除性 |
| 收稿时间： | 2021/9/4 0:00:00 |
| 修稿时间： | 2022/5/7 0:00:00 |
More Divisibility Properties for Binomial Coefficients |
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| Jiaqi XIAO,Yuqing HE,Pingzhi YUAN.More Divisibility Properties for Binomial Coefficients[J].Journal of Mathematical Research with Applications,2022,42(6):561-579. |
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| Authors: | Jiaqi XIAO Yuqing HE Pingzhi YUAN |
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| Institution: | School of Mathematics, South China Normal University, Guangdong 510631, P. R. China |
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| Abstract: | Let $a$, $b$ and $n$ be positive integers with $a>b$, we prove the following divisibility property: For all positive integers $n$, we have $$(2bn+1)(2bn+3)(2bn+5){2bn\choose bn}\Big|15(a-b)(3a-b)(5a-b)(5a-3b){2an \choose an}{an\choose bn},$$ which extends the result of Yang. And for all positive integers $n$, we show the following divisibility properties: $$(6n+1){4n\choose n}\Big|{12n\choose 6n}{2n\choose n},\ (12n+1){5n\choose n}\Big|{15n\choose 3n}{3n-1\choose n-1},$$ $$(18n+1){12n\choose 9n}{8n\choose 2n}\Big| {24n\choose 18n}{4n\choose 2n}{6n\choose 3n}.$$ Other more similar divisibility properties are given also. |
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| Keywords: | binomial coefficients $p$-adic order divisibility property |
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