The density of primes dividing a particular non-linear recurrence sequence
Volume 175 / 2016
Acta Arithmetica 175 (2016), 71-100
MSC: Primary 11G05; Secondary 11F80.
DOI: 10.4064/aa8265-4-2016
Published online: 3 August 2016
Abstract
Define the ECHO sequence $\{b_n\}$ recursively by $(b_0,b_1,b_2,b_3)=(1,1,2,1)$ and for $n\geq 4$, $$ b_n=\begin{cases} \dfrac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}} & \text{if }n\not\equiv 0\pmod 3,\\ \dfrac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}} &\text{if }n\equiv 0\pmod 3.\end{cases} $$ We relate $\{b_n\}$ to the coordinates of points on the elliptic curve $E:y^2+y=x^3-3x+4$. We use Galois representations attached to $E$ to prove that the density of primes dividing a term in this sequence is equal to $\frac{179}{336}$. Furthermore, we describe an infinite family of elliptic curves whose Galois images match those of $E$.
