Analytically determine all possible integer solutions (x,y) of each of the following equations:

(A) √(x+ √(x+ √( x +......+ √(x + √x))))) = y

(B) √(x+ √(x+ √(x + √(x ......+ √(x + √(2x))))) = y, with x < 180,000

The square root symbol in the each of the above relationships is repeated 2007 times.

in order to find a solution (x,y) for part B I will create a series Zn defined as such

Z(n)=Sqrt[x+Sqrt[Z(n-1)]]

Z(0)=2x

now it is plain to see that we have Z(2006)=y

now it would make things very simple if we could have that Z(n)=Z(n-1) or 2x=Sqrt[x+Sqrt[2x]]  get rid of those nasty sqrt's and you have

4x^2=x(4x-1)

so x=0 is an obvious solution

devide thru by x and you get

4x=4x-1 obviously no other solutions exist

So now all there is left to do is prove that there is no such x that allows Z(2006) to be an integer

Posted by Daniel on 2007-02-20 01:00:08