plastic constant

Given the equation $P^3=P+1$, solve for $P$. The only solution in real numbers is $P=\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\sqrt[3]{\frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}=\frac{\sqrt[3]{12(9+\sqrt{69})}+\sqrt[3]{12(9-\sqrt{69})}}{6}\approx 1.3247179572447$, and $P$ is the plastic constant, also known as the silver number.

Another way to calculate the plastic constant is $\frac{P(n)}{P(n-1)}$, where $P(n)$ is the $n^{th}$ term of either the Padovan sequence or the Perrin sequence. For about $n>20$ the approximation is adequate for all practical purposes.

Title	plastic constant
Canonical name	PlasticConstant
Date of creation	2013-03-22 16:10:39
Last modified on	2013-03-22 16:10:39
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	8
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11B39
Synonym	plastic number
Synonym	silver number
Synonym	silver constant