If $p>0$ and$ q>0$, And $p^2-q^2$ is a prime number, then $p-q$ is...? Please help

Question

The answer is part $d)$none of the above, How do i proceed with such questions?

Answer 1

Note that $$p^2-q^2=(p-q)(p+q)$$

From the definition of prime numbers, it follows that $p-q=1$.

If $p-q \neq 1$, then $p-q$ is a number that divides $p^2-q^2$ that is not $1$ or $p^2-q^2$.

Answer 2

given Since $p,q$ are integers and $p^2-q^2=(p-q)(p+q)$ is prime, it means that we got integer factors of primes number. And since we know that the only factors of prime are $1$ and prime only therefore we are force to believe that $p-q=+1$ or $-1$ which leaves no other choice as integrs are positive