Let "x" and "y" be the solution. Then   (after multiplication both sides by  ) you will get


 =    --->       ( Divide both sides by  )

 = ,   or         ( after introducing the new variable t =  )

 = t   --->    = t   --->   =   --->  t =   --->  t =   --->  t = .


Then from one side,  =  is a rational number.
From the other side,   is an irrational number.

Contradiction.

The contradiction proves the original statement.

The proof is completed.

There are 2 solutions in integers.  [3 if you count 
interchanging the x and y values on one of them as 
a separate solution.]





Let , and 





Multiply through by 

















The smallest integer for "a" that will give
a positive value for b is a=3

If , 

If , 

If ,   <--not an integer.

If ,  

That's all the solutions which are integers,
because  is a decreasing function
from b=3 approaching b=2 asymptotically.

Since , and 

Substituting a=3, b=6 , and 

Substituting a=4, b=4 , and 

Substituting a=6, b=3 , and 

There are 2 solutions, (x,y) = (45,180), (x,y) = (80,80)
There are 3 if you count (x,y) = (180,45) as a separate solution
from the first one.

-------
Checking:

1. x = 45, y = 180











That shows that it is a solution.
  
x = 80, y = 80









That shows that it is also a solution.

Edwin