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

TL;DR

This paper investigates the distribution of primes dividing terms in the Somos-5 sequence, establishing that their density is exactly 5087/10752 by linking the sequence's properties to an elliptic curve and Galois representations.

Contribution

It connects the arithmetic of the Somos-5 sequence to elliptic curve theory and Galois representations to precisely compute the prime divisors' density.

Findings

Prime divisors of the Somos-5 sequence have density 5087/10752.

The sequence's properties are related to the elliptic curve y^2 + xy = x^3 + x^2 - 2x.

Galois representations are used to analyze the distribution of prime divisors.

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 $E : y^{2} + xy = x^{3} + x^{2} - 2x$ 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 $\frac{5087}{10752}$.