Primes of form $N^2 - 2$ where $N$ is odd and greater than $1$

This should be an open problem! Your problem is essentially prove that $N^2-2$ is prime infinitely often. This would be a special case of Bunyakovsky conjecture. If $N^2-2$ were not prime infinitely often, this would prove a counterexample to the conjecture (roughly speaking, it does not satisfy all the necessary conditions).