# Pythagorean Triangle/Examples/3059-8580-9109

## Example of Primitive Pythagorean Triangle

The triangle whose sides are of length $3059$, $8580$ and $9109$ is a primitive Pythagorean triangle.

File:3059-8580-9109.png

It has generator $\tuple {78, 55}$.

## Proof

We have:

 $\ds 78^2 - 55^2$ $=$ $\ds 6084 - 3025$ $\ds$ $=$ $\ds 3059$

 $\ds 2 \times 78 \times 55$ $=$ $\ds 8580$

 $\ds 78^2 + 55^2$ $=$ $\ds 6084 + 3025$ $\ds$ $=$ $\ds 9109$

 $\ds 3059^2 + 8580^2$ $=$ $\ds 9 \, 357 \, 481 + 73 \, 616 \, 400$ $\ds$ $=$ $\ds 82 \, 973 \, 881$ $\ds$ $=$ $\ds 9109^2$

It follows by Pythagoras's Theorem that $3059$, $8580$ and $9109$ form a Pythagorean triple.

We have that:

 $\ds 3059$ $=$ $\ds 7 \times 19 \times 23$ $\ds 8580$ $=$ $\ds 2^2 \times 3 \times 5 \times 11 \times 13$

It is seen that $3059$ and $8580$ share no prime factors.

That is, $3059$ and $8580$ are coprime.

Hence, by definition, $3059$, $8580$ and $9109$ form a primitive Pythagorean triple.

The result follows by definition of a primitive Pythagorean triangle.

$\blacksquare$