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arXiv 2011.033262020-11-05math.NT

On members of Lucas sequences which are either products of factorials or product of middle binomial coefficients and Catalan numbers

Shanta Laishram

备注：: arXiv admin note: text overlap with arXiv:2006.01756
发表信息：: Research in Number Theory 2021
DOI：: 10.1007/s40993-021-00257-x

英文摘要：

Let $\{U_n\}_{n\geq 0}$ be a Lucas sequence. Then the equation $$|U_n|=m_1!m_2!\cdots m_k!$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12\}$. Further the equation $$|U_n|=D_{m_1}D_{m_2}\cdots D_{m_k}, \qquad D_{m_i}\in \{B_{m_i}, C_{m_i}\}$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12, 16\}$. Here $B_m$ is the middle binomial coefficient $\binom{2m}{m}$ and $C_m$ is the Catalan number $\frac{1}{m+1}\binom{2m}{m}$.

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