In this page there is a necessary and sufficient test given for testing Pythagorean triples:
A simpler, more powerful test is, (by naming the even leg a): $(c − a)$ and $\large\frac{(c − b)}{2}$ are both perfect squares. This is both necessary and sufficient for the triple to be a PT.
Using this here,we can write $a = 444,b=333,c=555$,which means $111$ and $\frac{222}{2}=111$ must be perfect squares but it is not.Hence,that will not work.
Is there any necessary and sufficient condition that will work for every and any (other than summing up the squares and checking for perfect squares?
NOTE: $333,444,555$ are Pythagorean triples as $3\times111,4\times111,5\times111$ for $3,4,5$ is a Pythagorean triple.