The primes are :
The prime gaps are thus: 3-2,5-3,7-5,11-7,13-11,17-13,... 1,2,2,4,2,4,...
An example subsequence of the prime gaps sequence is: 2,4,2
This subsequence begins at 5, since the initial 2 is the gap between 7 and 5. Shock and horror, though, this sequence is not able to be uniquely located, since the identical sequence also begins at 11.
(As an aside to the main question, it would be very interesting to consider counting, estimating, or bounding, the multiplicities of any subsequence. For instance, how often does 2,4,2 occur? This is obviously a generalization of the 'twin prime' conjecture to arbitrary contiguous subsequences of the prime gaps, and almost certainly must have been considered. Anyone know a reference to this?)
Then, given an arbitrary contiguous subsequence of the prime gaps sequence (i.e. such that each term of the subsequence is precisely the difference between a prime and the immediate next prime), I am interested in 2 questions:
- Is it ever possible to identify, for any such subsequence, a prime that begins it? That is, is it possible to locate an arbitrary contiguous gap subsequence, without knowing which primes it expresses the differences of?
- If one considers some requirements, obviously length of the subsequence must be a factor of how easy it is, or with how much confidence we can locate a sequence. Therefore, what is the minimum length required to locate an arbitrary subsequence, assuming it is possible to locate it?
Finally, a justification : why should this be interesting? A measure of the length of the subsequence required to locate the primes it expresses the differences of should give a ballpark estimate of the information contained in the differences of the primes, and how specific these are to the primes themselves. This is very interesting. It would also, obviously, be interesting to be convinced of the fact that it is impossible to ever locate any subsequence (with the exception of any one beginning at 2 which provides the unique singlet difference 1). This would be equally fascinating. Either way, a result of argument would serve as an interesting jumping off point for my (and perhaps others) further thinking about these things.
Please, I am not interested in quibbling over precise definitions of all terms in this question, I am interested in gaining greater understanding of the ideas expressed here. I believe they are of sufficient clarity that further translation into rigorous and precise formulations is not required for readers to ascertain exactly what is being sought, thanks.
In case you are in any doubt, consider the words of W.S Anglin 'Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.'. So let us keep it loose, instead of couched in the familiar but somewhat suffocating cotton-wool of rigour, for now, thanks.