论文标题
对于$ q $ - arithmetic进程的最小常见倍数的非平凡有效下限
Nontrivial effective lower bounds for the least common multiple of a $q$-arithmetic progression
论文作者
论文摘要
本文致力于建立序列$ {（u_n）} _ {n \ in \ Mathbb {n}} $的连续术语中最不常见的倍数的非平凡有效下限$ \ mathrm {gcd}（u_0，r）= \ mathrm {gcd}（u_1，q）= 1 $。对于这样的顺序，我们表明，对于所有正整数$ n $，我们都有$ \ mathrm {lcm} \ {u_1，u_2，u_2，\ dots，u_n \} \ geq c_1 \ geq c_1 \ cdot c_2^n \ cdot q^n \ cdot q^{仅取决于$ q，r $和$ u_0 $。这可以被认为是作者（2005年）和Hong and Feng（2006年）在算术上已经获得的$ Q $ analog。
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.
