Prime divisibility in a prime square bandtwidth

Question

I am seeking your support for proving (or fail) formally the following homework:

Let $p_j\in\Bbb P$ a prime, then any $q\in\Bbb N$ within the interval $p_j<q<p_j^2$ is prime, if and only if all:

$$p_{i}\nmid q$$ with $1\le i<j$

There should be a simple sieving argument behind of this that I can not fish.

I hope I got it now correct.

Your help is welcome

Answer 1

The condition of $q \nmid p_{j-1}$ is absurd, since $q > p_{j-1}$.

So let's assume that we flip the order.

Even then, the statement is still not true. For example, consider $p_4 = 7$, then 8 is a valid counter example.

---

I believe that the statement you want it:

$q$ is a prime if and only if $ p_i \nmid q$ for all values of $i<j$.

This is a true statement that is easy to show.

Hint: If $q$ is not a prime number, then it must have a prime factor that is less than $\sqrt{q} < p_j$. If $q$ is a prime, then clearly no smaller number divides it.