I've been number crunching irreducible Pythagorean triples and this pattern came up: the difference between the hypotenuse and the larger leg seems to always be n² or 2n² for some integer n. Moreover, every integer of the form n² or 2n² is the difference between hypotenuse and the larger leg for some irreducible Pythagorean triple. Is there a simple proof for that?
There is a result listed on Wikipedia that looks kinda, sorta related: that the area of a Pythagorean triangle can not be the square or twice the square of a natural number.
EDIT: Actually, the statements are correct only for odd n² (but also by any 2n² as stated) as John Omielan demonstrated below.