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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.
Highlights
The recurrence Fn = Fn−1 + Fn−2, with initial values F0 = 0 and F1 = 1 yields the celebrated Fibonacci numbers
Its zeroes are much studied in literature: we refer to Martin (2004), Miles (1960), Miller (1971), Wolfram (1998), and Zhu and Grossman (2009), where it is proved that the unique positive root tends to 2, as k → ∞
A lemma follows regarding the roots of its characteristic polynomial
Summary
For k prove that its unique positive root k tends to the golden ratio We derive bounds on k in terms of Fibonacci numbers. 1+ 2 is the positive root of the characteristic polynomial Its zeroes are much studied in literature: we refer to Martin (2004), Miles (1960), Miller (1971), Wolfram (1998), and Zhu and Grossman (2009), where it is proved that the unique positive root tends to 2, as k → ∞.
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