Definition:Pythagorean Triangle
Definition
A Pythagorean triangle is a right triangle whose sides all have lengths which are integers.
Examples
$3-4-5$ Triangle
The triangle whose sides are of length $3$, $4$ and $5$ is a primitive Pythagorean triangle.
$6-8-10$ Triangle
The triangle whose sides are of length $6$, $8$ and $10$ is a Pythagorean triangle.
This is not a primitive Pythagorean triangle.
$5-12-13$ Triangle
The triangle whose sides are of length $5$, $12$ and $13$ is a primitive Pythagorean triangle.
$7-24-25$ Triangle
The triangle whose sides are of length $7$, $24$ and $25$ is a primitive Pythagorean triangle.
$693-1924-2045$ Triangle
The triangle whose sides are of length $693$, $1924$ and $2045$ is a primitive Pythagorean triangle.
$4485-5852-7373$ Triangle
The triangle whose sides are of length $4485$, $5852$ and $7373$ is a primitive Pythagorean triangle.
$3059-8580-9109$ Triangle
The triangle whose sides are of length $3059$, $8580$ and $9109$ is a primitive Pythagorean triangle.
$1380-19 \, 019-19 \, 069$ Triangle
The triangle whose sides are of length $1380$, $19 \, 019$ and $19 \, 069$ is a primitive Pythagorean triangle.
Also see
Source of Name
This entry was named for Pythagoras of Samos.
Historical Note
Pythagorean triangles were known to the ancient Babylonians in about $2000$ BCE.
The cuneiform tablet Plimpton $\mathit { 322 }$ list $15$ sets of Pythagorean triples.
It is also clear that the author of that tablet was also familiar with the fact, proved in Solutions of Pythagorean Equation, that the numbers $2 p q$, $p^2 - q^2$ and $p^2 + q^2$ always make a Pythagorean triangle.
It is likely that the ancient Greeks, and in particular Pythagoras or one of his disciples, obtained this result from the Babylonians.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$