A 120 degree triple is a solution, , in positive integers to the 120 degree triangle equation
If additionally ,and are pairwise relatively prime then is a primitive 120 degree triple.
is a primitive 120 degree triple if and only if there exists relative prime integers and , and such that
For , the Fibonacci number is given by where . The first few are .
Some notation:
.
is a 120 degree triple. And if then it's a primitive triple. That is
where each side of the triangle is relatively prime to each of the other two sides.
Hence and are relatively prime.
Let and . Then . So, we have
and |
Let , then , , , and . So
and |
.
This works for generalized Fibonacci numbers also. That is, choose any two positive integers and , then obtain integers , and thusly,
Example: If and then , and . Therefore
Construct equilateral triangles on each of the shorter legs of the triangle in Figure (2) creating the two triangles and as shown in figure (3). Thus,
and |