
Pythagorean triples
Find all Pythagorean triples (not necessary primitive) with one of the three integers equal to 16. (i.e. x, y, z in N* with x^2 + y^2 = z^2 and 16 in {x, y, z}).
I know how to find the answer for this problem but don't know how to show my answer in a proper way. The answer is {(16, 12, 20), (12, 16, 20), (16, 30, 34), (30, 16, 34), (16, 63, 65), (63, 16, 65)}. Please help me to show all the steps. Thank you so much.

 Case I : ( the case without solutions if you mean )
Note that: Just because
So: implies that and with
So:
And repeat the process until we get: which is a contradiction since !
 Case II : Without loss of generality ( since we can exchange for freely) take
We have: and factorise
So we must have (since 2 is prime) : for and clearly since
Sum the 2 equations to get: , hence is even, and so
Thus it follows: (*) (solve the system)
Now take and (*) will produce solutions (note that hence can take 3 values )
Remember to consider the case which is totally analogous, all you have to do is to exchange for