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arXiv 1706.03350math.NT
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$(1^{\ell}+q^{\ell})(2^{\ell}+q^{\ell})\cdots (n^{\ell}+q^{\ell})$中的强大数字

Powerful numbers in $(1^{\ell}+q^{\ell})(2^{\ell}+q^{\ell})\cdots (n^{\ell}+q^{\ell})$

Quan-Hui Yang, Qing-Qing Zhao

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7 pages
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摘要：

令$q$为正整数。最近，牛和刘证明了，如果$n\ge \max\{q,1198-q\}$，那么产品$(1^3+q^3)(2^3+q^3)\cdots (n^3+q^3)$不是一个强大的数字。本文证明了：（1）对任意奇素数幂$\ell$和$n\ge \max\{q,11-q\}$，积$(1^{\ell}+q^{\ell})(2^{\ell}+q^{\ell})\cdots (n^{\ell}+q^{\ell})$不是幂数;（2）对任意正奇整数$\ell$，存在整数$N_{q,\ell}$，使得对任意正整数$n\ge N_{q,\ell}$，积$(1^{\ell}+q^{\ell})(2^{\ell}+q^{\ell})\cdots (n^{\ell}+q^{\ell})$不是幂数.

英文摘要：

Let $q$ be a positive integer. Recently, Niu and Liu proved that if $n\ge \max\{q,1198-q\}$, then the product $(1^3+q^3)(2^3+q^3)\cdots (n^3+q^3)$ is not a powerful number. In this note, we prove that (i) for any odd prime power $\ell$ and $n\ge \max\{q,11-q\}$, the product $(1^{\ell}+q^{\ell})(2^{\ell}+q^{\ell})\cdots (n^{\ell}+q^{\ell})$ is not a powerful number; (2) for any positive odd integer $\ell$, there exists an integer $N_{q,\ell}$ such that for any positive integer $n\ge N_{q,\ell}$, the product $(1^{\ell}+q^{\ell})(2^{\ell}+q^{\ell})\cdots (n^{\ell}+q^{\ell})$ is not a powerful number.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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