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Generating functions is not really the right name. I would say "parameterization."

These formulas were known to the Babylonians. The simple proof goes as follows: assume WLOG that a, b, c are relatively prime. Since a, b, c cannot all have the same parity, WLOG b, c have different parity. Then a^2 = (c + b)(c - b) where the factors on the RHS are odd and relatively prime, so they must both be squares.

It's equivalent to showing that abc is divisible by 3, 4, 5. This is straightforward if you know that squares are congruent to 0, 1 mod 3, congruent to 0, 1 mod 4, and congruent to 0, 1, 4 mod 5 because 1 + 1 != 1 mod 3 or mod 4 and similarly for 5.