morphic number

The golden ratioMathworldPlanetmathφ=1+52satisfies the equations

{φ+1=φ2,φ-1=φ-1 (1)

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

{P+1=P3,P-1=P-4 (2)

Here, the latter equation is justified by

P5-P4-1(P3-P-1= 0)(P2-P+1)

when this is divided by P4.

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

{x+1=xm,x-1=x-n (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.


  • 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