Peter H van der Kamp | Searcharxiv

arXiv2015

Somos-4 and Somos-5 are arithmetic divisibility sequences

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.

math.NT↗

arXiv2014

From integrable equations to Laurent recurrences

Based on a recursive factorisation technique we show how integrable difference equations give rise to recurrences which possess the Laurent property. We derive non-autonomous Somos-$k$ sequences, with $k=4,5$, whose coefficients are periodic functions with period 8 for $k=4$, and period 7 for $k=5$, and which possess the Laurent property. We also apply our method to the DTKQ-$N$ equation, with $N=2,3$, and derive Laurent recurrences with $N+2$ terms, of order $N+3$. In the case $N=3$ the recurrence has periodic coefficients with period 8. We demonstrate that recursive factorisation also provides a proof of the Laurent property.

nlin.SI↗