Fall 2016, problem 24
The Fibonacci sequence is defined by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Show that for any natural number $m$, there exists an index $k$ such that $a_k^4-a_k-2$ is divisible by $m$.
The Fibonacci sequence is defined by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Show that for any natural number $m$, there exists an index $k$ such that $a_k^4-a_k-2$ is divisible by $m$.