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.