1. Prove that there is no largest prime.

Proof: Assume by contradiction that there is a largest prime $\displaystyle n $. Then its divisors are $\displaystyle 1 $ and $\displaystyle n $. Then how would I continue?

2. If $\displaystyle n $ is a positive integer, prove that the algebraic identity $\displaystyle a^{n} - b^{n} = (a-b) \sum_{0}^{n-1} a^{k} b^{n-1-k} $. So use induction on $\displaystyle n $?

3. If $\displaystyle 2^{n}-1 $ is prime, prove that $\displaystyle n $ is prime. Then $\displaystyle 2^{n} $ is not prime. But this does not imply that $\displaystyle n $ is prime.