I am looking for references regarding the following statement.
For any two natural numbers x and y there must be a prime k-tuple (a, b, ...) corresponding to x consecutive primes (n+a, n+b, ...) for at least y different n.
I am not familiar with much of the literature and would like to know whether it is a known result. I offer my idea of a proof here:
It is detailed in my answer under this (my) question. The idea/structure of my (attempt at a) proof is given under Approach. The proof is written in terms of prime patterns rather than k-tuples. The term prime pattern is defined and linked to k-tuple in the section Glossary.
I would much appreciate any kind of answer. I am an amateur mathematician with fairly little formal training and limited, eclectic knowledge of the literature, but have been reading and thinking about prime numbers for 20 years.
Since the first commenter (Gerhard Paseman) has mentioned the twin primes conjecture and asks for a list of consequences, these are my own thoughts on both.
If the above is a theorem, then my understanding is that:
1.) there must either be patterns (k-tuples) of any size and inifnite abundance or be infinitely many different patterns (k-tuples) of any size and a given abundance
2.) the result also holds for non-overlapping patterns (k-tuples). That is, for any x and y one can find k-tuples (a, b, ...) with arbitrarily many different n such that none of the resulting primes (n+a, n+b, ...) for one n is equal to any prime given by one of the other n. (To be clear: I am not saying there are no two n with overlap, just that one can find arbitrarily many n without any overlap)
Regarding the twin primes conjecture, the Polignac conjecture, Dickson's conjecture, the prime k-tuples conjecture and Zhang's theorem:
I don't think this result implies any of these. It is obviously far more elementary than one would expect a proof of any of these to be. Unlike them, it does not state that there are infinitely abundant k-tuples, only that there are some k-tuples of any finite abundance. It allows only an either-or-conclusion as to what is inifinitely abundant (see Consequences).