On the least common multiple of shifted powers

Abstract

Let $a \geq 2$ be an integer. We prove that for every periodic sequence $(s_n)_{n \geq 1}$ in $\{-1, +1\}$ there exists an effectively computable rational number $C_\mathbf{s} > 0$ such that \begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim C_\mathbf{s} \cdot \frac{\log a}{\pi^2} \cdot n^2 , \end{equation*} as $n \to +\infty$, where $\operatorname{lcm}$ denotes the least common multiple. Furthermore, we show that if $(s_n)_{n \geq 1}$ is a sequence of independent and uniformly distributed random variables in $\{-1, +1\}$ then \begin{equation*} \log\operatorname{lcm}(a + s_1, a^2 + s_2, \dots, a^n + s_n) \sim 6 \operatorname{Li}_2\!\big(\tfrac1{2}\big) \cdot \frac{\log a}{\pi^2} \cdot n^2 , \end{equation*} with probability $1 - o(1)$, as $n \to +\infty$, where $\operatorname{Li}_2$ is the dilogarithm function.