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 $(9,12,15)=3\times (3,4,5)$, to be generated.

Πηγή: theoremoftheday