# Pythagorean Triangle/Examples/4485-5852-7373

## Example of Primitive Pythagorean Triangle

The triangle whose sides are of length $4485$, $5852$ and $7373$ is a primitive Pythagorean triangle.

File:4485-5852-7373.png

It has generator $\tuple {77, 38}$.

## Proof

We have:

 $\ds 77^2 - 38^2$ $=$ $\ds 5929 - 1444$ $\ds$ $=$ $\ds 4485$

 $\ds 2 \times 77 \times 38$ $=$ $\ds 5852$

 $\ds 77^2 + 38^2$ $=$ $\ds 5929 + 1444$ $\ds$ $=$ $\ds 7373$

Hence:

 $\ds 4485^2 + 5852^2$ $=$ $\ds 20 \, 115 \, 225 + 34 \, 245 \, 904$ $\ds$ $=$ $\ds 54 \, 361 \, 129$ $\ds$ $=$ $\ds 7373^2$

It follows by Pythagoras's Theorem that $4485$, $5852$ and $7373$ form a Pythagorean triple.

We have that:

 $\ds 4485$ $=$ $\ds 3 \times 5 \times 13 \times 23$ $\ds 5852$ $=$ $\ds 2^2 \times 7 \times 11 \times 19$

It is seen that $4485$ and $5852$ share no prime factors.

That is, $4485$ and $5852$ are coprime.

Hence, by definition, $693$, $1924$ and $2045$ form a primitive Pythagorean triple.

The result follows by definition of a primitive Pythagorean triangle.

$\blacksquare$