Greatest common divisor and Fibonacci numbers

Many people know a very basic fact about Fibonacci numbers : \[ gcd(F_n, F_{n+1}) = 1 \] In other words: \(F_n\) and \(F_{n+1}\) are coprime There is a more generic form: \[ gcd(F_n, F_m) = F_{gcd(n, m)} \] The first proposition follows from this, as \(gcd(n, n+1) = 1\) . Some time ago Johannes and I discovered another fact if n is coprime to 2 and 3: \[ n \bmod{6} \in \{ 1, 5 \} \Rightarrow…