The density of primes dividing a particular non-linear recurrence sequence
Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, Jeremy Rouse
Acta Arithmetica 175 (2016), 71-100
MSC: Primary 11G05; Secondary 11F80.
DOI: 10.4064/aa8265-4-2016
Opublikowany online: 3 August 2016
Streszczenie
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 $E$.
Autorzy
- Alexi Block GormanDepartment of Mathematics
Wellesley College
Wellesley, MA 02481, U.S.A.
e-mail
- Tyler GenaoDepartment of Mathematical Sciences
Florida Atlantic University
Boca Raton, FL 33431, U.S.A.
e-mail
- Heesu HwangDepartment of Mathematics
Princeton University
Princeton, NJ 08544, U.S.A.
e-mail
- Noam KantorDepartment of Mathematics
Emory University
Atlanta, GA 30322, U.S.A.
e-mail
- Sarah ParsonsDepartment of Mathematics and Statistics
Wake Forest University
Winston-Salem, NC 27109, U.S.A.
e-mail
- Jeremy RouseDepartment of Mathematics and Statistics
Wake Forest University
Winston-Salem, NC 27109, U.S.A.
e-mail