Suppose a and b are positive integers. We all know that aČ+2ab+bČ is a perfect square. Give an example where also aČ+ab+bČ is a perfect square. How many such examples exist?
(In reply to re: Complete Solution
An inadvertent omission had occurred in my post giving the complete solution to the problem under reference.
While the given formula a= K(x + y)(x-3y) and b = 4Kxy does generate the desired pairs (a,b) corresponding to positive integers x and y; the magnitude of K is POSITIVE RATIONAL NUMBER AND THEREFORE WOULD ALSO BE INCLUSIVE OF INTEGRAL VALUES. I would shortly be amending the texts constituting my original post.
Regarding your query, it would be observed that substituting
x=5; y=1 and K = 1/4 will yield the desired pair (a,b) = (3,5)