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

TL;DR

This paper establishes new effective lower bounds for the least common multiple of consecutive terms in a specific $q$-arithmetic progression, extending previous results for standard arithmetic progressions with explicit bounds depending on sequence parameters.

Contribution

It introduces nontrivial lower bounds for the LCM of $q$-arithmetic progressions, generalizing earlier bounds for arithmetic progressions with explicit constants.

Findings

Lower bounds involve exponential growth in $n$ and $q^{n^2/4}$.

Bounds depend explicitly on sequence parameters $q, r, u_0$.

Results extend classical bounds to $q$-analog sequences.

Abstract

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.