You are referring to the generalized proof of " Twin-Prime Conjecture ", by using the Prime Number Theorem ( PNT ).
Prime number theorem states that the density of primes is ruled by the law $\pi(n) \sim \large\frac{n}{log(n)}$ We can state this in a more precise form using Riemann's Li function, as $\pi(n)=Li(n)+O(ne^{-a\sqrt{Log(n)}})$ for some constant $a$.
we can compute that the probability of finding twin primes ( $a-b=2$ , but you are looking at $2k$ which will be an implication ) $a$ at $n + 1$ and $b$ at $n -1$ is about $\large\frac{2}{\rm{Log(n)}}$ , but you are writing about the case for some $k$ but here in this case I mentioned above its $k=1$ and the number of twin primes in the interval $n$ is about $\large\frac{2n}{\rm{Log(n)}^2}$.
So its clearly evident to say from the above things that
- The difference of two prime numbers $a$ and $b$ is an even number $2k$.
- The probability of finding primes $a$ at $n + k$ and $b$ at $n - k$ decreases as $n$ increases.
The prime number theorem states that the number of primes less than $n$ is asymptotic to $1/\rm{Log(n)}$. So if we choose a random integer $m$ from the interval $[1,n]$, then the probability that $m$ is prime is asymptotic to $1/\rm{Log(n)}$
So its crystal clear that one can apply the same thing to your conditions by adjusting the values and as the $n$ is in the denominator of probability its evident that they are inversely proportional.
I end here, and I seriously advice you to completely go through this and this.
But I find there are some more beautiful articles about this, but it takes some time for me to fish them out. I surely re-edit once if I find any of such things.
Thank you,
Yours truly,
Iyengar.