Decidability of prime gap sequences Is the following problem undecidable?

Given a sequence of $n$ gaps $d_1,d_2,...,d_n$, does there exist a sequence of $n+1$ primes $p_1,p_2,...,p_{n+1}$ such that $p_{i+1} - p_i = d_i$ ?

If not, is the problem still undecidable if we pick a fixed $n$, e.g. only one gap (n=1), or only two gaps (n=2).
With undecidable I mean that you cannot build a Turing machine that given the sequence $d_1,...,d_n$ as input always halts and gives the correct answer.
 A: This is probably unknown. The problem is that we know very little about the distribution of prime numbers. In general, decidability questions only become interesting when we "have a good grasp" about the object we are manipulating. It is easy to generate decidability questions which are only hard because of our limited knowledge. Here are a few examples:


*

*Given two rationals $a<b$, can one decide whether there a zero of Reimann's zeta function whose real part belongs to $(a,b)$?

*Given a finite sequence of digits $w$, can one decide whether $w$ appears in the decimal expansion of $\pi$?

*Given an integer $n$, can one decide whether the Collatz sequence starting at $n$ eventually reaches $1$? 


In all three cases, the answer is assumed to be positive for a trivial reason (in the first case, by simply checking whether $a<1/2<b$ or whether $a<-2n<b$ for some nonnegative integer $n$, in the second and third case by answering `yes' all the time) but proving this does not appear to be simpler than the general conjecture. I feel like your problem somewhat falls into this category. 
