Some small Pythagorean triples are
In each case, the hypotenuse has length one greater one of the legs. Let’s try to find all such triples.
Let the longer of the legs be . Then the hypotenuse is . The shorter leg must be an integer, .
In order to get a Pythagorean triple, must also be an integer. That means must be even, so must be odd. That happens whenever itself is odd.
How about Pythagorean triples where one of the legs is two less than the hypotenuse?
So here we must have a multiple of , which happens when is even.
How about triples where a leg is less than the hypotenuse?
So the numerator, needs to be a multiple of . Since the second term already is one, so must be the first. That means
for some integer . Plugging that back into the equation for gives
So if is even, there’s nothing to worry about. Otherwise, must be odd.
Want to find a Pythagorean triple that has a leg shorter than the hypotenuse? Then so we can choose any odd number for . How about ?
We have the Pythagorean triple .