# derivation of plastic number

The *plastic number* may be defined to be the limit of the ratio of two successive members (http://planetmath.org/Sequence^{}) of the Padovan sequence^{} or the Perrin sequence^{}, both of which obey the recurrence relation

${a}_{n}={a}_{n-3}+{a}_{n-2}.$ | (1) |

Supposing that such a limit

$P:=\underset{n\to \mathrm{\infty}}{lim}{\displaystyle \frac{{a}_{n+1}}{{a}_{n}}}$ | (2) |

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

$\frac{{a}_{n}}{{a}_{n-1}}}\cdot {\displaystyle \frac{{a}_{n-1}}{{a}_{n-2}}}={\displaystyle \frac{{a}_{n-3}}{{a}_{n-2}}}+1$ | (3) |

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

$$P\cdot P=\frac{1}{P}+1,$$ |

which is same as

${P}^{3}=P+1.$ | (4) |

Thinking the graphs of the equations $y={x}^{3}$ and $y=x+1$, it is clear that the cubic equation^{}

${x}^{3}-x-1=\mathrm{\hspace{0.33em}0}$ | (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

$$({u}^{3}+{v}^{3}-1)+(3uv-1)(u+v)=\mathrm{\hspace{0.33em}0}.$$ |

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

$\{\begin{array}{cc}{u}^{2}+{v}^{3}=\mathrm{\hspace{0.33em}1},\hfill & \\ uv=\frac{1}{3},\text{or}\mathit{\hspace{1em}}{u}^{3}{v}^{3}=\frac{1}{27}\hfill & \end{array}$ |

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

$${z}^{2}-z+\frac{1}{27}=\mathrm{\hspace{0.33em}0},$$ |

i.e.

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

and accordingly

$$u=\sqrt[3]{\frac{9+\sqrt{69}}{18}},v=\sqrt[3]{\frac{9-\sqrt{69}}{18}}.$$ |

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

$$x=\sqrt[3]{\frac{9+\sqrt{69}}{18}}+\sqrt[3]{\frac{9-\sqrt{69}}{18}},$$ |

or

$P={\displaystyle \frac{\sqrt[3]{12(9+\sqrt{69})}+\sqrt[3]{12(9-\sqrt{69})}}{6}}.$ | (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 |
---|---|

Canonical name | DerivationOfPlasticNumber |

Date of creation | 2013-03-22 19:09:41 |

Last modified on | 2013-03-22 19:09:41 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Derivation |

Classification | msc 11B39 |

Related topic | LimitRulesOfSequences |

Related topic | GoldenRatio |

Related topic | LimitRulesOfFunctions |