# Pythagorean Triangle/Examples/4485-5852-7373

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## Example of Primitive Pythagorean Triangle

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

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.

$\blacksquare$