Simple proof of the integrality of the Somos-5 sequence

TL;DR

This paper provides a straightforward proof that the Somos-5 sequence consists solely of natural numbers and that each term is coprime with its four predecessors, extending previous results with a simplified approach.

Contribution

It offers a simple, accessible proof of the integrality of the Somos-5 sequence and its coprimality properties, building on earlier proofs for Somos-4 and Somos-5 sequences.

Findings

Somos-5 sequence produces only natural numbers

Each term is coprime with its four predecessors

Simplified proof method based on earlier work

Abstract

In this paper, a simple proof that shows that the Somos-5 sequence produces only natural numbers is given. We also give simple proof that each $a_n$ in the Somos-5 sequence is relatively prime to its four predecessors. The manner in which the integrality is proven is taken from the 1992 proof for the integrality of the Somos-4 sequence by Janice. L. Malouf. The manner in which it is proven that each $a_n$ is relatively prime to its four predecessors is taken from a paper from Michael J. Crabb, who proved that each $a_n$ in the Somos-5 sequence is relatively prime to its two predecessors.