Question

A Pythagorean triple is said to be "primitive" if the greatest common divisor of a, b, and c is 1. Use trial and error to find a primitive Pythagorean triple other than (3, 4, 5).

Answer 1

5, 12, 13

Explanation:

Answer 2

(5, 12, 13)

Explanation:

Hello, please consider the following.

`(\forall (x,y) \in \mathbb{R}^2) \ \ (x^2-y^2)^2+(2xy)^2=(x^2+y^2)^2\\\\\text{You can develop both expressions to see that this is equal.}\\\\\text{After }(3,4,5)\text{ we can try the triple starting with }5\\\\\text{So, let's search x and y such that } 5=x^2-y^2=9-4=3^2-2^2\\\\\text{And, then, it comes}\\\\(3^2-2^2)^2+(2*3*2)^2=(3^2+2^2)^2\\\\<=> 5^2+12^2=13^2\\ \\\text{So, }(5,12,13)\text{ is a Pythagorean triple}`

Is it a primitive one?

5 = 1 * 5

12 = 1 * 2 * 2 * 3

13 = 1 * 13

So this is a primitive one.

Thanks