Pythagorean Triangle/Examples/1380-19,019-19,069

From ProofWiki
Jump to navigation Jump to search

Example of Primitive Pythagorean Triangle

The triangle whose sides are of length $1380$, $19 \, 019$ and $19 \, 069$ is a primitive Pythagorean triangle.


File:1380-19,019-19,069.png


It has generator $\left({138, 5}\right)$.


Proof

We have:

\(\ds 138^2 - 5^2\) \(=\) \(\ds 19 \, 044 - 25\)
\(\ds \) \(=\) \(\ds 19 \, 019\)


\(\ds 2 \times 138 \times 5\) \(=\) \(\ds 1380\)


\(\ds 138^2 + 5^2\) \(=\) \(\ds 19 \, 044 + 25\)
\(\ds \) \(=\) \(\ds 19 \, 069\)


\(\ds 1380^2 + 19 \, 019^2\) \(=\) \(\ds 1 \, 904 \, 400 + 361 \, 722 \, 361\)
\(\ds \) \(=\) \(\ds 363 \, 626 \, 761\)
\(\ds \) \(=\) \(\ds 19 \, 069^2\)

It follows by Pythagoras's Theorem that $1380$, $19 \, 019$ and $19 \, 069$ form a Pythagorean triple.


We have that:

\(\ds 1380\) \(=\) \(\ds 2^2 \times 3 \times 5 \times 23\)
\(\ds 19 \, 019\) \(=\) \(\ds 7 \times 11 \times 13 \times 19\)

It is seen that $1380$ and $19 \, 019$ share no prime factors.

That is, $1380$ and $19 \, 019$ are coprime.

Hence, by definition, $1380$, $19 \, 019$ and $19 \, 069$ form a primitive Pythagorean triple.

The result follows by definition of a primitive Pythagorean triangle.

$\blacksquare$