A challenging problem on prime uncertainty interval

I have a very challenging problem to solve, seeking for good advice; I have to make an intro in the first part and then comming to the problem.

Theorem (1): In an interval between a prime $p$ and its square, $p<n<p^2$, all composite $n\in\Bbb N$ are divisble by at least one prime $p^-<p$.

Proof: Suppose that there exists a composite number that does not have a prime divisor less than $p$. Then the number in question must be greater than $(p+2)(p+2)=(p+2)^2$, contradiction. So done.

It follows that any natural number $n\in\Bbb N$ in the interval$p_j<n<p_j^2$ which is not divisible by all $p^-<p_j$ must be a prime.

The limit $p^2$ can be even set higher at a cost of uncertainty whether numbers harvested in a broader interval woud be then a prime or not.

Can anyone help with an approach or a type of uncertainty function for the probability of pointing out a prime (and not a composite) when extending the upper limit to an alternative higher level?

(1) Theorem to be cited Vaseghi 2013

• $(p_{j+1})^2$, I think, and not $p_{j+1} p_{j+2}$. Also, I don't understand why you would need to know whether the larger numbers are prime or not. It sounds like you want the following: Given that the number is composite, the probability that its least prime factor is less than $p_j$. +1 for well-written question. – Yoni Rozenshein May 26 '13 at 9:38