The infinite continued fraction: x + 1/(x+ 1/(x + 1/(x+ ....))) is denoted by C(x).

Determine all possible pair(s)(g, h) of positive integers that satisfy this equation:

  2            1
-----  -   ------ = 1
C(g)       C(h)

from previous post we have
C(x)=(x+sqrt(x^2+4))/2
so for x>=1 we have C(x)>=1
So for the equation to have a solution we need
2/C(g)>1 or
C(g)<2
(g+sqrt(g^2+4))/2<2
g+sqrt(g^2+4)<4
g^2+4<(g-4)^2
g^2+4<g^2-8g+16
8g<12
g<12/8=3/2=1.5
thus only possible positive integer value for g is g=1
if g=1 then C(g)=(1+sqrt(5))/2
putting this into the equation and solving for C(h) we have
C(h)=(4+sqrt(20))/2
and thus h=4 is the only solution when g=1 thus we have the only solution as (1,4)

Posted by Daniel on 2009-09-10 16:06:22