Prove that there exists a unique prime number of the form $n^2 − 1$, where *n* is an integer that is greater than or equal to 2.

Question

Prove that there exists a unique prime number of the form $n^2-1$, where n is an integer that is greater than or equal to 2.

I know I can factor $n^2-1 = (n-1)(n+1)$. What are my next steps?

Answer 1

Existence : $2^2 - 1 = 3$

Uniqueness : Suppose we have a number $n > 2$ with prime number of the form $n^2 - 1$. Then since $n^2 - 1 = (n+1)\cdot (n-1)$, we have $(n+1) > 1$ and $(n-1) > 1$, which means that both of these are factors that are not $1$ and thus $n^2-1$ is not prime.