📂Number Theory

Pythagorean Triple

Definition ¹

A set of three natural numbers $(a,b,c)$ that satisfies $a^2 + b^2 = c^2$ is called a Pythagorean triple. If the three natural numbers have no common factor, they are called a Primitive Pythagorean Triple.

Description

For convenience, let’s call the numbers included in a Pythagorean triple Pythagorean numbers.

Examples of Pythagorean triples, as everyone well knows, include $(3, 4, 5)$ and $(5, 12, 13)$ among others. The main focus is often on the Primitive Pythagorean triples, commonly abbreviated as PPTs. Assuming that the Pythagorean numbers have a common factor of $g$, resulting in $$ a=gA \\ b=gB \\ c=gC $$, then, because $$ \begin{align*} & a^2 + b^2 = c^2 \\ \iff & (gA)^2 + (gB)^2 = (gC)^2 \\ \iff & A^2 + B^2 = C^2 \end{align*} $$, considering all these possibilities is meaningless.