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