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arXiv 2008.10294math.NT
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$q$ -等差数列最小公倍数的非平凡有效下界

Nontrivial effective lower bounds for the least common multiple of a $q$-arithmetic progression

Bakir Farhi

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15 pages
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摘要：

本文研究了序列${(u_n)}_{n \in \mathbb{N}}$的连续项的最小公倍数的非平凡有效下界，其中$q , r$为正整数，$u_0$为非负整数，使得$\mathrm{gcd}(u_0 , r) = \mathrm{gcd}(u_1 , q) = 1$ .对于这样的序列，我们表明，对于所有的正整数$n$，我们有$\mathrm{lcm}\{u_1 , u_2 , \dots , u_n\} \geq c_1 \cdot c_2^n \cdot q^{\frac{n^2}{4}}$，其中$c_1$和$c_2$是正常数，只依赖于$q , r$和$u_0$。这可以被认为是作者（2005年）和Hong和Feng（2006年）已经获得的算术级数的下界的$q$模拟。

英文摘要：

This paper is devoted to establish nontrivial effective lower bounds for the least common multiple of consecutive terms of a sequence ${(u_n)}_{n \in \mathbb{N}}$ whose general term has the form $u_n = r {[n]}_q + u_0$, where $q , r$ are positive integers and $u_0$ is a non-negative integer such that $\mathrm{gcd}(u_0 , r) = \mathrm{gcd}(u_1 , q) = 1$. For such a sequence, we show that for all positive integer $n$, we have $\mathrm{lcm}\{u_1 , u_2 , \dots , u_n\} \geq c_1 \cdot c_2^n \cdot q^{\frac{n^2}{4}}$, where $c_1$ and $c_2$ are positive constants depending only on $q , r$ and $u_0$. This can be considered as a $q$-analog of the lower bounds already obtained by the author (in 2005) and by Hong and Feng (in 2006) for the arithmetic progressions.

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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