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

Alexi Block Gorman ¹   ; Tyler Genao ²   ; Heesu Hwang ³   ; Noam Kantor ⁴   ; Sarah Parsons ⁵   ; Jeremy Rouse ⁵

¹ Department of Mathematics Wellesley College Wellesley, MA 02481, U.S.A.
² Department of Mathematical Sciences Florida Atlantic University Boca Raton, FL 33431, U.S.A.
³ Department of Mathematics Princeton University Princeton, NJ 08544, U.S.A.
⁴ Department of Mathematics Emory University Atlanta, GA 30322, U.S.A.
⁵ Department of Mathematics and Statistics Wake Forest University Winston-Salem, NC 27109, U.S.A.

Acta Arithmetica, Tome 175 (2016) no. 1, pp. 71-100

Résumé

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 $E$.

Détail

DOI : 10.4064/aa8265-4-2016

Keywords: define echo sequence recursively geq begin cases dfrac n n b n n text equiv pmod dfrac n n n n text equiv pmod end cases relate coordinates points elliptic curve galois representations attached nbsp prove density primes dividing term sequence equal frac furthermore describe infinite family elliptic curves whose galois images match those