jmc

$25^2+72^2=5^4+4\cdot 6^4$, and we can use the Sophie Germain Identity on that to get $$25^2+72^2=(5^2+2\cdot 6^2+2\cdot 5\cdot 6)(5^2+2\cdot 6^2-2\cdot 5\cdot 6)=157\cdot 37.$$ $\boxed{157}$ is the largest prime factor.