A growth model based on the arithmetic $Z$-game

TL;DR

This paper introduces a novel growth model based on a numerical rule involving gcd, generating fractal matrices with intriguing geometric and evolutionary properties, including solitons and edge behaviors, with connections to prime factorizations.

Contribution

It presents a new self-governing growth model using the $Z$-rule, analyzes the resulting fractal matrices, and solves a problem related to the matrix's western edge.

Findings

The matrix exhibits fractal structures and solitons.

The shape and properties of a related matrix are characterized.

A problem posed by Sloane is solved, and a conjecture is supported.

Abstract

We present an evolutionary self-governing model based on the numerical atomic rule $Z(a,b)=ab/\gcd(a,b)^2$, for $a,b$ positive integers. Starting with a sequence of numbers, the initial generation $Gin$, a new sequence is obtained by applying the $Z$-rule to any neighbor terms. Likewise, applying repeatedly the same procedure to the newest generation, an entire matrix $T_{Gin}$ is generated. Most often, this matrix, which is the recorder of the whole process, shows a fractal aspect and has intriguing properties. If $Gin$ is the sequence of positive integers, in the associated matrix remarkable are the distinguished geometrical figures called the $Z$-solitons and the sinuous evolution of the size of numbers on the western edge. We observe that $T_{\mathbb{N}^*}$ is close to the analogue free of solitons matrix generated from an initial generation in which each natural number is replaced by its largest divisor that is a product of distinct primes. We describe the shape and the properties of this new matrix. N. J. A. Sloane raised a few interesting problems regarding the western edge of the matrix $T_{\mathbb{N}^*}$. We solve one of them and present arguments for a precise conjecture on another.