Exhibit five primes of the form $n^2-2$, $n \in \mathbb{N}$

Question

It has been conjectured that there are infinitely many primes in the form $n^2-2$. Exhibit five such primes.

I'm so confused what the problem is asking. Do I just need to find examples or an actual proof?

Answer 1

"Exhibit" just means find, so we are looking for $5$ prime numbers that are two less than a square.

First note than an even number squared minus $2$ will always be even, except in the case of $2^2 - 2$, so let us start there and proceed checking the odd numbers.

Let's list the first few square numbers and subtract $2$ from each and see what we get

\begin{align*} 2^2 &\to 4-2 = 2 \qquad \text{ prime}\\ 3^2 &\to 9-2 = 7 \qquad \text{ prime}\\ 5^2 &\to 25 - 2 = 23 \qquad \text{ prime}\\ 7^2 &\to 49 - 2 = 47 \qquad \text{ prime}\\ 9^2 &\to 81 - 2 = 79 \qquad \text{ prime} \end{align*} and we have found $5$ such primes!