Squaring The Triangle

TRIVIA: According to Fermat , the smallest such triple has sides $a = 4565486027761, b = 1061652293520$ and $c = 4687298610289$ .

Here we have $a+b = 2372159^2$ and $c = 2165017^2$ , so $\sqrt{a+b} - \sqrt{c} = 207142$ .

It can be factorized as $207142 = 2 \cdot 13 \cdot 31 \cdot 257$ , thus our desired sum is $2+13+31+257 = \boxed{303.}$

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An original solution to this problem would obviously require some code computing.