Find all positive integers (x,y) that satisfy y=√(x+x√x)
Find all positive integers (x,y) that satisfy y=√(x+x√(x+x√x))
y= V(x+xVx)
or, y^2 = x+xVx
or, (y^2-x)^2 =x^3 =p^6(say)
Then, y^2= p^3+p^2 = p^2 * (p+1)
Accordingly, p+1 must be a perfect square
Let p+1= q^2(say)
or, p=q^2-1
Then, y^2= (q^2 -1)^2 * q^2
or, y= q(q^2-1) and x= (q^2-1)^2 as x=p^2
Consequently, (x,y)= {(q^2-1)^2, q(q^2-1)} is a parametric formula giving all possible solutions to the given puzzle.
AS A CHECK
y= q(q^2-1)
or, (xVx) = (q^2-1)^3
or, x+xVx=(q^2-1)^2 +(q^2 -1)^3 = q^2 * (q^2 -1)^2
or,V(x+ xVx) = q(q^2 -1) = y
Puzzle Solution: Part A
