On the index of appearance of a Lucas sequence

Carlo Sanna

Published 2022-12-12Version 1

Let $\mathbf{u} = (u_n)_{n \geq 0}$ be a Lucas sequence, that is, a sequence of integers satisfying $u_0 = 0$, $u_1 = 1$, and $u_n = a_1 u_{n - 1} + a_2 u_{n - 2}$ for every integer $n \geq 2$, where $a_1$ and $a_2$ are fixed nonzero integers. For each prime number $p$ with $p \nmid 2a_2D_{\mathbf{u}}$, where $D_{\mathbf{u}} := a_1^2 + 4a_2$, let $\rho_{\mathbf{u}}(p)$ be the rank of appearance of $p$ in $\mathbf{u}$, that is, the smallest positive integer $k$ such that $p \mid u_k$. It is well known that $\rho_{\mathbf{u}}(p)$ exists and that $p \equiv \big(D_{\mathbf{u}} \mid p \big) \pmod {\rho_{\mathbf{u}}(p)}$, where $\big(D_{\mathbf{u}} \mid p \big)$ is the Legendre symbol. Define the index of appearance of $p$ in $\mathbf{u}$ as $\iota_{\mathbf{u}}(p) := \left(p - \big(D_{\mathbf{u}} \mid p \big)\right) / \rho_{\mathbf{u}}(p)$. For each positive integer $t$ and for every $x > 0$, let $\mathcal{P}_{\mathbf{u}}(t, x)$ be the set of prime numbers $p$ such that $p \leq x$, $p \nmid 2a_2 D_{\mathbf{u}}$, and $\iota_{\mathbf{u}}(p) = t$. Under the Generalized Riemann Hypothesis, and under some mild assumptions on $\mathbf{u}$, we prove that \begin{equation*} \#\mathcal{P}_{\mathbf{u}}(t, x) = A\, F_{\mathbf{u}}(t) \, G_{\mathbf{u}}(t) \, \frac{x}{\log x} + O_{\mathbf{u}}\!\left(\frac{x}{(\log x)^2} + \frac{x \log (2\log x)}{\varphi(t) (\log x)^2}\right) , \end{equation*} for all positive integers $t$ and for all $x > t^3$, where $A$ is the Artin constant, $F_{\mathbf{u}}(\cdot)$ is a multiplicative function, and $G_{\mathbf{u}}(\cdot)$ is a periodic function (both these functions are effectively computable in terms of $\mathbf{u}$). Furthermore, we provide some explicit examples and numerical data.