Somos-4 and Somos-5 are arithmetic divisibility sequences

TL;DR

This paper proves that multiples of prime powers in the Somos-4 sequence are evenly spaced and extends these divisibility properties to generalized Somos-4 and Somos-5 sequences, revealing new arithmetic structure.

Contribution

Provides an elementary proof of a conjecture on prime power multiples in Somos-4 and establishes divisibility relations in generalized Somos sequences.

Findings

Multiples of prime powers in Somos-4 are equally spaced.

Divisibility of generalized Somos-4 sequence polynomials by shifted indices.

Similar divisibility properties hold for generalized Somos-5 sequence.

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 $\tau_{n+2}\tau_{n-2}=\alpha\tau_{n+1}\tau_{n-1}+\beta\tau_n^2$ and initial values $\tau_1=\tau_2=\tau_3=\tau_4=1$, that the polynomial $\tau_n(\alpha,\beta)$ is a divisor of $\tau_{n+k(2n-5)}(\alpha,\beta)$ for all $n,k\in\mathbb Z$ and establish a similar result for the generalized Somos-5 sequence.