I haven't read the paper you posted yet, but just to clear up a common confusion, Pythagorean triples are not the Pythagorean theorem. The theorem is a logical

I haven't read the paper you posted yet, but just to clear up a common confusion, Pythagorean triples are not the Pythagorean theorem. The theorem is a logical statement about all right triangles, and it has a proof that the statement holds. Pythagorean triples are specific instantiations of the relation for some known triangles and probably would have served as evidence that the statement was even provable. Historically we probably had triples long before we had a proof of the theorem, just like many of the theorems proved by Euclid were probably already known as rules of thumb. Compare with an open problem today, like the Riemann hypothesis.