Pythagorean Triple
A Pythagorean triple is a set of three positive integers that satisfy the equation $a^2 + b^2 = c^2$.
Formula
a^2 + b^2 = c^2 \text{ with } a, b, c \text{ positive integers}
Definition
A Pythagorean triple is a group of three positive integers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$, the most famous being $3$, $4$, $5$. Primitive triples (no common factor) can be generated by $a = m^2 - n^2$, $b = 2mn$, $c = m^2 + n^2$ for integers $m > n > 0$ with $\gcd(m, n) = 1$ and $m, n$ not both odd. They correspond to rational points on the unit circle $x^2 + y^2 = 1$ via $(a/c, b/c)$, and the rational parametrization $t = \tan(\theta/2)$ connects triples to Moebius transformations.
Example
The triples $3$-$4$-$5$, $5$-$12$-$13$, and $8$-$15$-$17$ are all Pythagorean triples, and multiples also work, such as $6$-$8$-$10$. Using $m = 3$, $n = 2$: $a = 5$, $b = 12$, $c = 13$. The rational point $(3/5, 4/5)$ on $x^2 + y^2 = 1$ corresponds to the triple $3$-$4$-$5$, via stereographic projection from $(-1, 0)$.
Key Insight
Pythagorean triples give right triangles with whole-number sides, so calculations come out exact with no rounding, and there are infinitely many primitive triples, with every odd number greater than $1$ part of at least one. The set of all Pythagorean triples is in bijection with the rational points on the unit circle minus $(-1, 0)$, corresponding to the rational projective line $P^1(\mathbb{Q})$, a prototype for modern arithmetic geometry; Fermat's Last Theorem (proven by Wiles in 1995) shows no analogous triples exist for exponents greater than $2$.