📂Number Theory

Pythagorean Triples

Definition ¹

An ordered triple $(a,b,c)$ of three natural numbers satisfying $a^2 + b^2 = c^2$ is called a Pythagorean triple. If the three natural numbers have no common divisor, it is called a primitive Pytahgoras Triple.

Explanation

For convenience, let us call the numbers contained in a Pythagorean triple Pythagorean numbers.

As examples of Pythagorean triples, there are the well-known ones such as $(3, 4, 5)$ and $(5, 12, 13)$. The main object of interest is the primitive Pythagorean triple, commonly written with the abbreviation PPT. Suppose the Pythagorean numbers have a common divisor $g$, so that they are expressed as

$$ a=gA \\ b=gB \\ c=gC $$

Then, since

$$ \begin{align*} & a^2 + b^2 = c^2 \\ \iff & (gA)^2 + (gB)^2 = (gC)^2 \\ \iff & A^2 + B^2 = C^2 \end{align*} $$

holds, there is no point in considering all these possibilities.