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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?

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    2 is not divisible by 3, even though $2\times 2 -1$ and $2\times 2 +1$ are prime2012-09-24

4 Answers 4