Suppose $p,q$ are two primes at even distance $k$. Must there necessarily exist a different pair $p',q'$ composed of entirely different numbers such that $p'$ and $q'$ are also at distance $k$?


Edit: None of the primes need to be consecutive.



If there are two primes at an even distance $k$ is there another disjoint pair of primes also at distance $k$?
It is conjectured that for *every* even number there are *infinitely* many pairs of primes at that distance. 


Thus conjecturally the answer is: yes.


However, this is not known though there was huge progress around this problem recently by Zhang, Maynard and others. 


The methods used to establish existence of primes at a certain distance is not typically helped by knowing one pair at that distance. 


To wit, note that the Twin Prime conjecture, infinitely many primes at distance 2, is open, while obviously there is are instances of pairs at this distance known. But this does not help much to get new ones. 


It is an open problem if there is an even distance where there is exactly one pair of primes at that distance. It seems extremely unlikely, but it cannot be at present excluded. So, what you ask is an open problem.


