The plastic number and its generalized polynomial

TL;DR

This paper explores the roots of a family of generalized polynomials, showing their positive roots approach the golden ratio and establishing bounds using Fibonacci numbers, thus extending the concept of the plastic number.

Contribution

It generalizes the study of the plastic number by analyzing roots of a broader class of polynomials and connecting their behavior to the golden ratio and Fibonacci bounds.

Findings

The unique positive root $ ho$ of $X^3 - X - 1$ is approximately 1.3247.

The roots $\lambda_k$ of the generalized polynomial tend to the golden ratio as $k$ increases.

Bounds on $\lambda_k$ are derived using Fibonacci numbers.

Abstract

The polynomial $X^{3}-X-1$ has a unique positive root known as plastic number, which is denoted by $\rho$ and is approximately equal to $1.32471795$. In this note we study the zeroes of the generalized polynomial $X^{k}-\sum_{j=0}^{k-2}X^{j}$ for $k\geq 3$ and prove that its unique positive root $\lambda_{k}$ tends to the golden ratio $\phi=\frac{1+\sqrt{5}}{2}$ as $k \to \infty$. We also derive bounds on $\lambda_{k}$ in terms of Fibonacci numbers.