Five digit pair

Let $a = m^2$ and $b = n^2$ . Therefore, we have:

$\begin{aligned} b - a & = n^2 - m^2 = (n+m)(n-m) = 11111 \end{aligned}$

Since $11111 = 41 \times 271$ has only two prime factors. It means that:

$\begin{cases} n+m = 271 \\ n-m = 41 \end{cases} \quad \Rightarrow \begin{cases} 2n = 312 & \Rightarrow n = 156 \\ m = 156 - 41 & \Rightarrow m = 115 \end{cases}$

$\Rightarrow \begin{cases} b = n^2 = 156^2 = 24336 \\ a = m^2 = 115^2 = 13225 \end{cases} \quad \Rightarrow a + b = \boxed{37561}$

37561 is correct as it can be broken down to two numbers ( 13225;24336) which satisfy the property.