IL-GILLIS 2015 Problem 7

The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$ . Let $p\geq3$ be a prime number.

[list=a] [] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$ . [] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$ . [/list]