morphic number
The golden ratio^{} $\phi =\frac{1+\sqrt{5}}{2}$ satisfies the equations
$\{\begin{array}{cc}\phi +1={\phi}^{2},\hfill & \\ \phi -1={\phi}^{-1}\hfill & \end{array}$ | (1) |
from which the latter is obained from the former by dividing by $\phi $. There is a pair of equations satisfied by the plastic number $P$:
$\{\begin{array}{cc}P+1={P}^{3},\hfill & \\ P-1={P}^{-4}\hfill & \end{array}$ | (2) |
Here, the latter equation is justified by
$${P}^{5}-{P}^{4}-1\equiv (\underset{=\mathrm{\hspace{0.33em}0}}{\underset{\u23df}{{P}^{3}-P-1}})({P}^{2}-P+1)$$ |
when this is divided by ${P}^{4}$.
An algebraic integer^{} is called a morphic number, iff it satisfies a pair of equations
$\{\begin{array}{cc}x+1={x}^{m},\hfill & \\ x-1={x}^{-n}\hfill & \end{array}$ | (3) |
for some positive integers $m$ and $n$.
Accordingly, the golden ratio and the plastic number are morphic numbers. It can be shown that there are no other real morphic numbers greater than 1.
References
- 1 J. Aarts, R. Fokkink, G. Kruijtzer: Morphic numbers. – Nieuw Archief voor Wiskunde 5/2 (2001).
Title | morphic number |
---|---|
Canonical name | MorphicNumber |
Date of creation | 2013-03-22 19:09:51 |
Last modified on | 2013-03-22 19:09:51 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 6 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 11B39 |