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.