## Further Symmetry In Primes

Two disjoint (?) sets of primes can be generated using two distinct rules:

The first set `A`

uses primes:

p' = {x | 3p - 2}; p is prime

Is the output of `(3p - 2)`

always a prime?

The second set `B`

uses positive integers:

p" = {x | P(6k - 1)}; P(.) is prime

Using primes `k = p`

that make true both conditions for set `A`

and set `B`

results in this interesting sequence of integers (all primes):

3,5,7,23,43,47,53,67,103, ... , ?

Now, the ratio `p"/p'`

produces the plot below, suggesting that as `p=k`

approaches infinity, there will be a `p"`

prime that is `2p`

(???)

The largest prime I have tested is:

p = 1159337; p' = 3478009; p" = 6956021

with ratio: `2.00000086256246`