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arXiv · 1505.00194

Somos-4 and Somos-5 are arithmetic divisibility sequences

Peter H van der Kamp
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Abstract

We provide an elementary proof to a conjecture by Robinson that multiples of (powers of) primes in the Somos-4 sequence are equally spaced. We also show, almost as a corollary, for the generalised Somos-4 sequence defined by $τ_{n+2}τ_{n-2}=ατ_{n+1}τ_{n-1}+βτ_n^2$ and initial values $τ_1=τ_2=τ_3=τ_4=1$, that the polynomial $τ_n(α,β)$ is a divisor of $τ_{n+k(2n-5)}(α,β)$ for all $n,k\in\mathbb Z$ and establish a similar result for the generalized Somos-5 sequence.

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Peter H van der Kamp. 2015-05-01. Somos-4 and Somos-5 are arithmetic divisibility sequences. https://arxiv.org/abs/1505.00194

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