The plastic number and its generalized polynomial

Cogent Mathematics

Vasileios Iliopoulos¹ 

[1] University of Essex;

关键词: Fibonacci;    golden ratio;    plastic number;   

DOI  :  10.1080/23311835.2015.1023123

来源: DOAJ

【 摘 要 】

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 generalised polynomial $ X^{k}-\sum _{j=0}^{k-2}X^{j} $, for $ k \ge 3 $, and prove that its unique positive root $ \lambda _{k} $ tends to the golden ratio $ \phi =\frac{1+\sqrt{5}}{2} $ as $ k \rightarrow \infty $. We also derive bounds on $ \lambda _{k} $ in terms of Fibonacci numbers.

【 授权许可】

Unknown