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I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible.

Yitang Zhang breakthrough result established that there is a number $k<7\times10^7$ for which there are infinite pairs of primes $(p,p+k)$. This $k$ was later improved. My question is: Can I find a bound $X$ and a constant $k$ such that for a given number $n$, I may find n disjoint pairs of prime numbers $(p_1,p_2),(p_3,p_4),\dots,(p_{2n-1},p_{2n})$, where:

$$ p_1 < p_2 <\dots< p_{2n-1} < p_{2n} < X \\\text{and}\\ p_{2i} \leq p_{2i-1}+k, i=1,\dots,n $$

That is, a set of $n$ prime pairs with a gap less or equal to some constant k between the element of each pair and all primes are not greater than $X$.

I would like to find $X$ as a function of $n$ (hopefully bounded by a polynomial of $n$).