Pythagorean Triangle/Examples/693-1924-2045

Example of Primitive Pythagorean Triangle

The triangle whose sides are of length $693$, $1924$ and $2045$ is a primitive Pythagorean triangle.

Proof

\(\ds 693^2 + 1924^2\)

\(=\)

\(\ds 480 \, 249 + 3 \, 701 \, 776\)

\(\ds \)

\(=\)

\(\ds 4 \, 182 \, 025\)

\(\ds \)

\(=\)

\(\ds 2045^2\)

It follows by Pythagoras's Theorem that $693$, $1924$ and $2045$ form a Pythagorean triple.

Note that $693$ and $1924$ are coprime.

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

The result follows by definition of a primitive Pythagorean triangle.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$