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

Davis, Bryant; Kotsonis, Rebecca; Rouse, Jeremy

doi:10.1090/bproc/26

The density of primes dividing a term in the Somos-5 sequence
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by Bryant Davis, Rebecca Kotsonis and Jeremy Rouse HTML | PDF

Proc. Amer. Math. Soc. Ser. B 4 (2017), 5-20

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 $\frac {5087}{10752}$.

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