Pythagorean Triangle/Examples/3059-8580-9109
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Example of Primitive Pythagorean Triangle
The triangle whose sides are of length $3059$, $8580$ and $9109$ is a primitive Pythagorean triangle.
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.
$\blacksquare$