Proof of divisibility by full induction

Question

How can I prove by full induction that $n^{2(p+1)}-1$ is divisible by $n+1$?

Answer 1

Hint:

• $n^2-1=(n+1)(n-1)\qquad$

• Consider $n^{2(p+1)}-n^{2p}$

Answer 2

Hint:

If $n^x\equiv1\bmod{n+1}$, and $n^2\equiv1\bmod{n+1}$, then $n^{x+2}\equiv1\bmod{n+1}$.