Newsgroups: sci.math
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Date: 1995/11/25
Subject: Re: Number sequences problem

rklei...@moe.coe.uga.edu (Ronen Kleiner M) asks:

> Take any four real numbers, a1,b1,c1,d1.
> Define an ("a sub n")=abs(a(n-1) - b(n-1)), bn=abs(b(n-1)-c(n-1)),
> cn=abs(c(n-1)-d(n-1)) and dn=abs(d(n-1)-a(n-1)).  ["abs"=absolute value].

and asks if iteration of this process eventually reaches {0,0,0,0}.
The answer is no.

The polynomial x^3 + x^2 + x - 1 has one real root, near 0.543689, or
exactly

  ( (3 sqrt(33)+17)^(1/3) - (3 sqrt(33)-17)^(1/3) - 1 ) / 3.

The sequence {x,1,(2x-1)/x,(1-x-x^2)/x} ~ {0.543689,1,0.160713,0.295598}
does not terminate.  It maps to

  {1-x, (1-x)/x, (2-3x-x^2)/x, (1+x)(2x-1)/x};

the longevity is not affected when we multiply each element by
x/(1-x), to get

  {x, 1, (2-3x-x^2)/(1-x), (1+x)(2x-1)/(1-x) }

which differs from the original sequence by

  {0, 0, (1-x-x^2-x^3)/(x(1-x)), -(1-x-x^2-x^3)/(x(1-x))}.

But these terms are all zero, by the definition of x.

I wonder if this is the only eigenvector of the system (up to rotation
and scaling)?  Are there higher-order cycles?  Is there any literature
on this process?

Dan
Hoey@AIC.NRL.Navy.Mil