The Theorem of the Day: Euclid’s Pythagorean Formula

For each Pythagorean triple $(a, b, c)$ (i.e. positive integers satisfying $a^2 + b^2 = c^2)$ there is a unique triple $(k, m, n)$ of positive integers, with $m > n$, and $m$ and $n$ coprime and of different parity, such that

$a = k(m^2 − n^2), b = 2kmn, c = k(m^2 + n^2)$.

Euclid’s formula appears in Book $10$ of his Elements where it is the content of Lemma $1$, used in the proof of Proposition $29$. The parameter k is not part of the original formula but is introduced to allow ‘imprimitive’ triples, such as $(9, 12, 15) = 3 × (3, 4, 5)$, to be generated.

Πηγή: theoremoftheday