Assuming Goldbach’s conjecture (that is, whenever ), let’s write .
I call the order of centrality of and say that is a -central number if and only if .
Now let’s consider and . Let and and . Zhang’s theorem implies , while Cramer’s conjecture implies . The Prime Number Theorem asserts that, on average, , so that is very likely to be equal to .
Let’s now define the following density: .
I formulate the following conjectures:
Symmetric Density Conjecture:
Increasing Density conjecture:
It may be worth noticing that these two conjectures are a consequence of a rather natural hypothesis of rotational invariance.
Would these last two conjectures imply Cramer’s conjecture?
By the way, I think that the fact that whenever is a potential typical primality radius of , then so is , implies symmetric density conjecture.
Moreover, the same kind of reasoning can be applied mutatis mutandis replacing the quantities by , by and by to get (as a consequence of Maynard’s work on bounded gaps between primes), and (as a consequence of both the fact that on average and that this last quantity is ), hence giving further evidence for the equality derived under a statistical heuristics in the previous article.