Greatest common divisor and Fibonacci numbers
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