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)

first off
C(x)=x+1/C(x)
C(x)^2-x*C(x)-1=0
C(x)=(x+sqrt(x^2+4))/2
Thus (2/C(g))-(1/C(h))=1 becomes
(4/(g+sqrt(g^2+4)))-(2/(h+sqrt(h^2+4)))=1
now using Mathematica to solve this for h I got
h=2(2+g-g^2+(g-1)sqrt(g^2+4))/(2g-3)
one obvious value that gives postive integer h is
g=1 because that drops out the radical with g-1=0
this give h=4 and thus we have the solution (1,4)
now to show that this is the only solution
we want h>0 thus
and once again using Mathematica is it easy to see this is only
true when g>0 and g<1.5 thus the only possible positive integer value of g that gives positive h is g=1

thus the only positive integer solution is (1,4)

Posted by Daniel on 2009-09-10 11:46:47