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First off, the terminology:

Primorials: the products of the first $n$ primes, written as $P_n \#$.

Reduced residue system modulo a positive integer $K$: Those numbers smaller than $K$ that are relatively prime with $K$.

$k$-tuple configuration: a pattern for a set $S$ of positive integers ordered by increasing value with differences $d_i$ between each consecutive pair of integers.

Background

We consider two sets of $k$-tuple configurations. The first are those $k$-tuple configurations that occur infinitely many times as prime $k$-tuples. The second set comes from those $k$-tuple configurations that theoretically occur in the reduced residue systems modulo infinitely many primorials.

The Question

Because all prime numbers must exist as members of the reduced residue system modulo at least one primorial, any $k$-tuple configuration that occurs infinitely many times among the primes will also occur in the reduced residue systems modulo infinitely many primorials.

Does this relationship extend the other way? If it can be shown that some $k$-tuple configuration occurs among the reduced residue systems modulo infinitely many primorials, would this mean that this same $k$-tuple configuration occurs infinitely many times among the primes?

Put differently, is there a bijection between the two sets of $k$-tuple configurations introduced above?

EDIT 1: As per the comments, it was noted that the $k$-tuple configurations I am talking about are synonymous with the idea of admissible prime k-tuples within the context of prime numbers. When extended over reduced residue systems modulo primorials, these $k$-tuples may not be all prime, meaning that they are not technically admissible, but they are otherwise identical to admissible $k$-tuples.

This observation makes it possible to state the question in terms of admissible prime $k$-tuples; if such a $k$-tuple configuration occurs infinitely many times among primes, it is guaranteed to have a counterpart that occurs in the reduced residue systems modulo infinitely many primorials. Does this relationship extend the other way? Given an extended-"admissible" $k$-tuple that occurs in the reduced residue systems modulo infinitely many primorials, will such a $k$-tuple also occur infintely often among the primes?

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  • $\begingroup$ Is this primorial residue system language just another way of describing the concept of "admissible" in the prime $k$-tuples conjecture? en.wikipedia.org/wiki/Prime_k-tuple $\endgroup$ – Erick Wong Mar 18 '14 at 8:03
  • $\begingroup$ Yes, thanks for pointing that out. I hadn't looked into the concept of admissible before, and that link makes it clear that the two ideas are one and the same. $\endgroup$ – Adam Mar 18 '14 at 17:47
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I'm not clear that I understand, but wouldn't half the primorial, plus or minus 2, 4, and 8, result in a centered co-prime sextuplet for infinitely many primorials, yet not necessarily result in an infinity of such prime sextuplets, though it clearly works for some of the smaller ones, e.g., primorials 30 and 210.

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