WakeUp wrote: Prove that for any odd positive integer $n$, $n^{12}-n^8-n^4+1$ is divisible by $2^9$. This quantity is $(n^4+1)(n^2+1)^2(n^2-1)^2$ and $2|n^4+1$ and $2^2|(n^2+1)^2$ and $2^6|(n^2-1)^2$