The question is to find out the number of ordered pairs $(a,b)$ where $a,b$ are non negative integers such that the equations $x^2-2ax+b=0$ and $x^2-2bx+a=0$ have integral roots.
For the roots to be integers the discriminant must be a perfect square.i got that $a^2-b$ and $b^2-a$ must be perfect squares.i am not able to proceed further.Any ideas?Thanks.
Hint: $a^2 - (a-1)^2 = 2a-1$. So if $a > 1$ then $a^2 - (a-1)^2 > a$. Now suppose $a \ge b > 1$. Can $a^2 - b$ be a perfect square ?