The Babylonians had written thousand of Pythagorean triples some with extremely large numbers. They must have had some way of generating them other than multiplying the 3,4,5 triple by a constant.
Here is a one way they might have used to find a Pythagorean triple.
First chose a prime number say 17.
17² = 289
divide 289 by 2 = 144.5
round 144.5 down to 144
round 144.5 up to 145
The new Pythagorean Triple is 17, 144, 145.
Here is an algebraic proof of the above: