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

@article{Gorman2015TheDO,
title={The density of primes dividing a particular non-linear recurrence sequence},
author={A. Gorman and Tyler Genao and Heesu Hwang and Noam Kantor and Sarah Parsons and Jeremy A. Rouse},
journal={arXiv: Number Theory},
year={2015}
}
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… Expand
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Arithmetic of quaternion origami
• Mathematics
• 2018
• 1
• Highly Influenced
• PDF

#### References

SHOWING 1-10 OF 14 REFERENCES
Surjectivity of mod 2n representations of elliptic curves
• Mathematics
• 2011
• 26
• Highly Influential
• PDF
Analytic Number Theory
• Mathematics
• 2004
• 2,423
• PDF
The arithmetic of elliptic curves
• J. Silverman
• Mathematics, Computer Science
• 1986
• 3,388
• PDF
Primes of the form x2 + ny2
• 475
• Highly Influential
• PDF
The Magma Algebra System I: The User Language
• Computer Science, Mathematics
• J. Symb. Comput.
• 1997
• 5,351
• PDF
The Magma algebra system. I
• The user language. J. Symbolic Comput.,
• 1997