# derivation of plastic number

 $\displaystyle a_{n}\;=\;a_{n-3}+a_{n-2}.$ (1)

Supposing that such a limit

 $\displaystyle P\;:=\;\lim_{n\to\infty}\frac{a_{n+1}}{a_{n}}$ (2)

exists (and is $\neq 0$), we first write (1) as

 $\displaystyle\frac{a_{n}}{a_{n-1}}\cdot\frac{a_{n-1}}{a_{n-2}}\;=\;\frac{a_{n-% 3}}{a_{n-2}}+1$ (3)

and then let $n\to\infty$.  It follows the limit equation

 $P\!\cdot\!P\;=\;\frac{1}{P}\!+\!1,$

which is same as

 $\displaystyle P^{3}\;=\;P\!+\!1.$ (4)

Thinking the graphs of the equations  $y=x^{3}$  and  $y=x\!+\!1$, it is clear that the cubic equation  $\displaystyle x^{3}\!-\!x\!-\!1\;=\;0$ (5)

has only one real root (http://planetmath.org/Equation), which is $P$.

For solving the plastic number from the cubic, substitute by Cardano (http://planetmath.org/CardanosFormulae) into (5) the sum  $x:=u\!+\!v$  of two auxiliary unknowns, when the equation may be written

 $(u^{3}\!+\!v^{3}\!-\!1)+(3uv\!-1)(u\!+\!v)\;=\;0.$

Then, as in the example of solving a cubic equation, $u$ and $v$ are determined such that the first two parentheses vanish:

 $\displaystyle\begin{cases}u^{2}\!+\!v^{3}\;=\;1,\\ uv\;=\;\frac{1}{3},\quad\mbox{or}\quad u^{3}v^{3}\;=\;\frac{1}{27}\end{cases}$

Thus $u^{3}$ and $v^{3}$ are the roots of the resolvent equation

 $z^{2}\!-\!z\!+\!\frac{1}{27}\;=\;0,$

i.e.

 $z\;=\;\frac{9\!\pm\!\sqrt{69}}{18}$

and accordingly

 $u\;=\;\sqrt{\frac{9\!+\!\sqrt{69}}{18}},\quad v\;=\;\sqrt{\frac{9\!-\!% \sqrt{69}}{18}}.$

Fixing that these the real values of the cube roots, we obtain the value of the plastic number in the form

 $x\;=\;\sqrt{\frac{9\!+\!\sqrt{69}}{18}}+\sqrt{\frac{9\!-\!\sqrt{69}}{18}},$

or

 $\displaystyle P\;=\;\frac{\sqrt{12(9\!+\!\sqrt{69})}+\sqrt{12(9\!-\!% \sqrt{69})}}{6}.$ (6)

By (5), $P$ is an algebraic integer  of degree (http://planetmath.org/DegreeOfAnAlgebraicNumber) 3 (and a unit of the ring of integers of the number field  $\mathbb{Q}(P)$).  For computing an approximate value of $P$, see e.g. nth root by Newton’s method.

Title derivation of plastic number DerivationOfPlasticNumber 2013-03-22 19:09:41 2013-03-22 19:09:41 pahio (2872) pahio (2872) 13 pahio (2872) Derivation msc 11B39 LimitRulesOfSequences GoldenRatio LimitRulesOfFunctions