Supercongruences on some binomial sums involving Lucas sequences

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• 作者：Guo-Shuai Mao ^{mg1421007@smail.nju.edu.cn} ; Hao Pan ; ^{haopan79@zoho.com}
• 关键词：Pell number ; Central binomial coefficient ; Congruence
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 April 2017
• 年：2017
• 卷：448
• 期：2
• 页码：1061-1078
• 全文大小：334 K
• 卷排序：448

文摘

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that∑k=0p−1(p−1k)(2kk)2Pk8k≡0(modp2) for any prime p≡7(mod8), where PkPk is the k-th Pell number. Further, we also propose three new congruences of the same type.