On Pythagorean Triples

Recall from basic geometry that the legs and hypotenuse of a right triangle are related to each other by the Pythagorean Theorem:

a² + b² = c²

where a and b are the lengths of the legs and c is the length of the hypotenuse.

There are certain three-number sets of that satisfy the Pythagorean Theorem in such a very exact fashion that they deserve to be called Pythagorean Triples. Let us enumerate some. The examples that follow are arranged in the order "a, b, c".

3, 4, 5

5, 12, 13

7, 24, 25

9, 40, 41

11, 60, 61

13, 84, 85

15, 112, 113

17, 144, 145

19, 180, 181

 In addition to these triples, their integer multiples are also Pythagorean triples. For instance,

18, 80, 82  are integer multiples of 9, 40, 41(by a factor of 2, in this case) and so are a Pythagorean triple.

Here's a challenge to you: 

1. Aside from satisfying the Pythagorean theorem, what other properties do these numbers have?

2. Given the properties you discovered, can you name other Pythagorean triples without actually plugging them in into the Pythagorean theorem?