Right Triangle's Proof A right triangle has all three sides integer lengths. One side has length 12. What are the possibilities for the lengths of the other two sides? Give a proof to show that you have found all possibilities.
EDIT: I figured out that there are a total of 4 combinations for a side with length 12.
$$a^2= c^2 -b^2$$
factoring the right side gives
$$a^2 = (c+b)(c-b)$$
so from there I just looked for values of c and b which would make the equation true. And I got: $(37,35), (13,5), (15,9), (20,16)$.
My only problem now is proving that these are all the possibilities. I have an intuition as to why that's true but I don't know how to go about a full proof. I'm guessing I need to make use of the fact that all sides are to be integer lengths and that $12$ itself cannot be equal to $c$.
I know that if I were to try values for values for $b$ and $c$ whose absolute value difference is greater than 8, then the equation would not hold true.
Ex:
$(37,35)$ has a difference of $2$ and works.
$(13,5)$ has a difference of $8$ and works.
$(15,9)$ has a difference of $6$ and works.
$(20,16)$ has a difference of $4$ and woks.
But if I were to pick any pair with an absolute difference greater than 8 it would not work. Can I just prove this is true with a couple of examples? Or do I need a full generic proof ? If so, how would I go about it?

Wolfram MathWorld gives the number of ways in which a number $n$ can be a leg (other than the hypotenuse) of a primitive or non-primitive right triangle.  For $n=12$, the number is $4$.  It also gives the number of ways in which a number $n$ can be the hypotenuse of a primitive or non-primitive right triangle.  For $n=12$, the number is $0$.  So the four triples you found are the only ones.
