The density of primes dividing a term in the Somos-5 sequence

Bryant Davis; Rebecca Kotsonis; Jeremy Rouse

paper

The density of primes dividing a term in the Somos-5 sequence

arXiv:1507.05896

Abstract

The Somos-5 sequence is defined by $a_{0} = a_{1} = a_{2} = a_{3} = a_{4} = 1$ and $a_{m} = \frac{a _{m - 1} a _{m - 4} + a _{m - 2} a _{m - 3}}{a _{m - 5}}$ for $m \geq 5$ . We relate the arithmetic of the Somos-5 sequence to the elliptic curve $E : y^{2} + x y = x^{3} + x^{2} - 2 x$ and use properties of Galois representations attached to $E$ to prove the density of primes $p$ dividing some term in the Somos-5 sequence is equal to $\frac{5087}{10752}$ .

arXiv comment15 pages