## Other methods for generating Pythagorean Triples

It turns out that these are Pythagorean triples as well. They are not mulitples of 3, 4, and 5

5, 12, 13

7, 24, 25

9, 40, 41

An interesting relationship can be seen.

- Take the first number of one of these triples. Let's use 5.
- Square it. = 25
- Subtract 1 from the square. = 24
- Half it. = 12
12 is the second number of the triple. One more than 12 = 13 is the 3rd number of the triple ... 5, 12, 13.

This can be shown as a formula for generating some Pythagorean Triples.

x , (x ² - 1) ÷ 2 , (x ² + 1) ÷ 2These formulas works well for odd initial number.

For instance, choose 11 .

- 11 ² =121.
- Subtract 1 and get 120.
- One-half of 120 is 60.
- One more than 60 is 61. Therefore,

- 11, 60, 61 is a Pythagorean Triple