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.



Decidability of prime gap sequences
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. 


