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

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 $\gcd (u_0,r)=\gcd (u_1,q)=1$. For such a sequence, we show that for all positive integer $n$, we have $\mathrm{lcm}\{u_1,u_2,\dots ,u_n\}\ge c_1\cdot c_2^n\cdot q^{\frac{n^2}{4}}$, 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.

arXiv comment15 pages

References in corpus (1)

• Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions