If a lottery is held with each card having the count of the number primes less than $x+h = x+x^{0.525(*)}$, $\pi(x+h)$ spots to mark, and the value of each spot at $\pi(p_n)$ is $p_n$ up to x afterward is only a guess. Rules: one mark spot, $s_1$ has to be less than $\sqrt x$ and $s_1$ along with the remaining mark spots must have a product such that the result is between

$$x < s_1^\alpha s_2^\beta...s_k^\omega < x+h$$

or show that $s_1$ is a prime $> x$. Only one product per card. The small winners are the first cardholder to show the smallest factor to fill the pigeonholes marked:

$$x < x+1, x+2, ... x+l, ... < x+h.$$

The large winner is the first cardholder which shows a $x+l$ does not have a factor and is prime. Jackpot winner, $J = x+l$, is the smallest value of $x+l$ of the large winners.

What is the minimal number of lottery cards you need to prove $J$ is prime?

How does this question differ from Cramér's white/black ball conjecture?

I am expecting the answer to be $l \le \lceil\left(\frac{x}{\pi(x)}\right)^2\rceil \forall x$ with $\lceil\cdot\rceil$ being the ceiling function, but do not hold me to it.

(*) This upper limit is placed on this only as to save search time and is a known bound to primes, x is the prior Jackpot winning number.


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