morphic number

The golden ratio  $\varphi=\frac{1+\sqrt{5}}{2}$satisfies the equations

 $\displaystyle\begin{cases}\varphi\!+\!1\;=\;\varphi^{2},\\ \varphi\!-\!1\;=\;\varphi^{-1}\end{cases}$ (1)

from which the latter is obained from the former by dividing by $\varphi$.  There is a pair of equations satisfied by the plastic number $P$:

 $\displaystyle\begin{cases}P\!+\!1\;=\;P^{3},\\ P\!-\!1\;=\;P^{-4}\end{cases}$ (2)

Here, the latter equation is justified by

 $P^{5}\!-\!P^{4}\!-\!1\;\equiv\;(\underbrace{P^{3}\!-\!P\!-\!1}_{=\;0})(P^{2}\!% -\!P\!+\!1)$

when this is divided by $P^{4}$.

An algebraic integer  is called a morphic number, iff it satisfies a pair of equations

 $\displaystyle\begin{cases}x\!+\!1\;=\;x^{m},\\ x\!-\!1\;=\;x^{-n}\end{cases}$ (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 MorphicNumber 2013-03-22 19:09:51 2013-03-22 19:09:51 pahio (2872) pahio (2872) 6 pahio (2872) Definition msc 11B39