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Somos-4 and Somos-5 are arithmetic divisibility sequences

2015/05/01 by van der Kamp, Peter H
#11B37 #11B83 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1505.00194

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 τn+2τn-2=ατn+1τn-1+βτn2 and initial values τ1234=1, that the polynomial τn(α,β) is a divisor of τn+k(2n-5)(α,β) for all n,k∈\mathbb Z and establish a similar result for the generalized Somos-5 sequence.

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