Sorry if this is trivial!
Consider the set 3, 6, 9, 15, 21, 30, 36, 51, 54, 69,
These are all such that $2x-1$ and $2x+1$ are both prime.
Why are they all divisible by 3?
And if say $yx-1$ and $yx+1$ are both prime (i.e. generated by the twin-prime-averages that are divisible by $y$), then do all these $x$'s have a common divisor?
Why?