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

$$P^5-P^4-1\equiv (\underset{=\mathrm{\hspace{0.33em}0}}{\underbrace{P^3-P-1}})(P^2-P+1)$$

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.

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