It's proved that if $P(n)$ is the sum of all different prime factors of $n$, then for all odd primes $p$ there is an inequality $$p\ge P(p^2-1)$$
An inequality holding for odd primes
Therefore there is an injection $j:\mathbb P \to \mathbb Q$, defined as $j(p)=\frac{p}{P(p^2-1)}$. Now to me the following conjecture seems very natural:
Given $\varepsilon >0$ there are two primes $p,q$ such that $|j(p)-j(q)|<\varepsilon$
The idea is to characterizing primes $p$ by the factorizations of their neighbors $p-1,p+1$.
A metric on the prime numbers?
$d(p,q)=|j(p)-j(q)|$ seems to fulfill the triangle inequality. Tested for all combinations of the 100 first primes.
I now realize that $d$ must be a metric because $j$ is an injection.
It is possible to define convergence of sequences of primes.