Pythagorean Triplets

We all know the basic triplets $3,4,5$ or $5,12,13$ or some even $7,24,25$ . However what is the general recipe to concoct all these.

Observe these two expansions $(a^2+b^2)^2 = a^4+2a^2b^2+b^4$ and

$(a^2-b^2)^2 = a^4-2a^2b^2+b^4$ . Subtracting we get $(a^2+b^2)^2-(a^2-b^2)^2 = 4a^2b^2$ which is same as $(a^2+b^2)^2-(a^2-b^2)^2 = (2ab)^2$ or

$(a^2+b^2)^2=(a^2-b^2)^2+(2ab)^2$ . Now choose different value of $a,b$ and get different triplets

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