TITLE: Is there a one-to-one function from the natural numbers to the primes?
QUESTION [12 upvotes]: Is there a function $f : \mathbb N \rightarrow \mathbb P$, where $\mathbb P = \{p \in \mathbb N \mid \ p$ is prime$\}$, such that $f$ is injective?
It is known that no such polynomial function exists.
Also, I wish to exclude functions defined as $f(n) = \text{the $n$th prime}$, which requires one to find primes in advance.

REPLY [2 votes]: We cannot completely rule out the possibility that a "simple" and "natural" function exists which very quickly enumerates a (possibly very thin) infinite subset of the primes.
For example it is not known if Catalan–Mersenne numbers are all primes, 2, 3, 7, 127, 170141183460469231731687303715884105727, ... (OEIS A007013). It would be a tremendous surprise if they were.
The practise of competing to find the largest known prime, and prizes like the EFF Cooperative Computing Awards, would become meaningless if someone could prove that all Catalan–Mersenne numbers are prime.
I have not heard of any mathematician believing that huge primes can be found that easily.