Unknown square

Here $8140613$ is a number of two squares summation.

First, if we consider that, the two square number is same. Then one of the square number should be $\frac{8140613}{2}=4070306.5\approx 2017.5^2$

After that, if we rewrite our equation, it stands

$\large b=\sqrt{8140613-a^2}$

Here, $a+b$ will be greatest, if $a$ or $b$ is closest to $2017.5...$

if we take $a=2017$ [Approximate to $2017.5.......$ ]

we get $b=2018$

So one answer is for $(a,b)=(2017,2018)$

$a+b=2017+2018=\boxed{\color{#20A900}4035}$

It's the greatest value..