By factoring $p^2 − 1$, we have $(p + 1)(p - 1)$.
I know that $p = 2$ which gives $3$ is the only solution.
However, how do I prove that $p = 2$ is the only integer which gives a prime?
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In $p^2-1=(p-1)(p+1)$ give a prime number only and only if p-1 is 1 so that p+1 will be the only prime factor. So n must be 2 and its the only one which can give us a prime(3). I better suggest you to think about $$n^2-m=p$$ where 'n' is an natural number and m<$n^2$ such that 'p' will be a prime number.