The density of primes dividing a particular non-linear recurrence sequence

TL;DR

This paper investigates the density of primes dividing terms of a specific non-linear recurrence sequence, linking it to elliptic curve properties and Galois representations, and computes an explicit prime density value.

Contribution

It establishes the prime density for the sequence using elliptic curve Galois representations and identifies a family of elliptic curves with matching Galois images.

Findings

Prime density of 9/336 for the sequence

Connection between the sequence and elliptic curve points

Identification of a family of elliptic curves with similar Galois images

Abstract

Define the sequence $\{b_n\}$ by $b_0=1,b_1=1, b_2=2,b_3=1$, and $$b_n=\begin{cases} \frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\not\equiv 0\pmod 3, \frac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\equiv 0\pmod 3. We relate this sequence $\{b_n\}$ 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 that of $E$.