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Nontrivial effective lower bounds for the least common multiple of a $q$-arithmetic progression (2008.10294v1)

Published 24 Aug 2020 in math.NT

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 $\mathrm{lcm}{u_1 , u_2 , \dots , u_n} \geq c_1 \cdot c_2n \cdot q{\frac{n2}{4}}$, 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.

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