Do there exists infinitely many primes that satisfy $p_a-p_b=k$

Question

I have read that Terence Tao proved that there exists infinitely many primes that satisfy $p_n-p_{n-1}\le246$ ($p_n$ denotes the $n^{th}$ prime)
I want to know whether it has been proven that there exists infinitely many primes that satisfy $p_a-p_{b}=k$, where $k$ is an integer ($k$ can be greater than 246)? I am specifically asking for equality (in the case of $p_n-p_{n-1}\le246$ there is only an inequality ). I am asking for a specific value of $k$.

Edit-1 The two primes need not be consecutive.
Edit-2
Suppose $p_a-p_{b}=k$ is true only for finite number of primes.
This would mean that there does not exists infinitely many primes that satisfy $p_n-p_{n-1}\le246$ (because it is not true for $k=1,2,….246$) leading to a contradiction.
This means that there exists infinitely many primes that satisfy $p_n-p_{n-1}= k~$ for some $k\le246$ even though we don't know the exact value of $k$ for which it is true .
Is this reasoning correct?

Answer 1

Your question is a special case of Dickson's conjecture which states

... for a finite set of linear forms $a_1 + b_1 n, a_2 + b_2 n, \ldots, a_k + b_k n$ with $b_i \ge 1$, there are infinitely many positive integers $n$ for which they are all prime, unless there is a congruence condition preventing this ...

There are $2$ linear forms involved in your question, with them being $a_1 = 0$ and $b_1 = 1$, i.e., $n$, plus $a_2 = k$ and $b_2 = 1$, i.e., $k + n$. Therefore, $n = p_b$ and $k + n = p_a$ gives what you're asking about, i.e.,

$$p_a - p_b = k \tag{1}\label{eq1A}$$

I'm quite certain no specific examples of Dickson's conjecture, like what you're asking about, have yet been proven.