Further Symmetry In Primes
2022-10-08 |
Two disjoint (?) sets of primes can be generated using two distinct rules:
The first set \(P'\) contains primes:
\(P' = {x | x = P(3p - 2)}\); \(p\) is prime and \(P(.)\) is prime
(Is \(x = (3p - 2)\) always a prime?)
The second set \(P"\) uses natural numbers \(n\) :
$$ P" = {x | x = P(6n - 1)}$$ where \(P(.)\) is prime
Using primes \(p = n\) results in a sequence of primes:
$$3,5,7,23,43,47,53,67,103, ... , ?$$
The ratio \(P"/P'\) produces the plot below
The largest prime tested is:
$$p = 1159337; p' = 3478009; p" = 6956021$$
with ratio: \(2.00000086256246\)