The density of primes dividing a particular non-linear recurrence sequence
type: Item
instance of: scholarly article
title: The density of primes dividing a particular non-linear recurrence sequence
zbMATH Open document ID: 1354.11043
author: Alexi Block Gorman
author: Tyler Genao
author: Heesu Hwang
author: Noam Kantor
author: Sarah Parsons
author: Jeremy Rouse
published in: Acta Arithmetica
publication date: 2016-11-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1508.02464
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 6654749
zbMATH Keywords: density of primes
zbMATH Keywords: elliptic curves
zbMATH Keywords: Galois representation
describes a project that uses: PARI/GP
describes a project that uses: Magma
describes a project that uses: SageMath
MaRDI profile type: Publication
OpenAlex ID: W2962792021
arXiv ID: 1508.02464
cites work: The Magma algebra system. I: The user language
cites work: The density of primes dividing a term in the Somos-5 sequence
cites work: Surjectivity of mod \(2^n\) representations of elliptic curves
cites work: Q4830109
cites work: Galois theory of iterated endomorphisms
cites work: The set of primes dividing the Lucas numbers has density 2/3
cites work: The Arithmetic of Elliptic Curves
DOI: 10.4064/AA8265-4-2016
review text: The original abstract is good, except for the second phrase which could be more precise, so here is the original abstract: ``Define the sequence \(\{b_n\}\) by \(b_0=1,b_1=1,b_2=2,b_3=1\), and NEWLINE\[NEWLINEb_n=\begin{cases} \frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\text{ if }n\not\equiv 0\pmod 3, \\ \frac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}}&\text{ if }n\equiv 0\pmod 3\end{cases}.NEWLINE\]NEWLINE 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 \(E\).''NEWLINENEWLINEI would change this phrase ``We relate this sequence \(\{b_n\}\) to the coordinates of points on the elliptic curve \(E:y^2+y=x^3-3x+4\).'' by: We relate this sequence \(\{b_n\}\) to the coordinates of points in the group generated by P(4,7) on the elliptic curve \(E:y^2+y=x^3-3x+4\).
reviewed by: Cyril Hugounenq
description: scientific article; zbMATH DE number 6654749