Pythagorean Triangle/Examples/1380-19,019-19,069
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.
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.
$\blacksquare$