# Greatest common divisor and Fibonacci numbers

Posted on 2011-07-21

Many people know a very basic fact about Fibonacci numbers:

In other words: and are coprime

There is a more generic form:

The first proposition follows from this, as .

Some time ago Johannes and i discovered another fact if n is coprime to 2 and 3:

Proofing it wasn’t very difficult; just use the identity to show the following:

Depending on the value of (only 2 possibilites: 1 and 3) you can choose a value for k which helps to easily simplify at least one sum or difference with the basic identity of the Fibonacci numbers