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) ÷ 2

These 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

Now you try it. What Pythagorean Triple begins with 15?