I found this in Beyond Infinity by Eugenia Cheng:
We can’t list all the numbers in the world because there are infinitely many. Nor can we list all the prime numbers in the world, but for a different reason – we don’t know what they are. (But if we did, we still wouldn't be able to list them as there are infinitely many.
I couldn't understand why we can't know prime numbers.
I understand that there are infinite prime numbers. But that can't be a reason to say that that we don't know what they are, right?
(If we had enough computing power, nothing is stoping us from calculating primes, isn't it?)
Because if that was the case, it could also be said that we don't know all numbers in the world as well, right?
So I guess there's something else.
What could it be?
With respect, I do not agree with the author here. There is a rigorous formal definition of what it means to be able to list a sequence $a_n$ of numbers, and it means you can write down a computer program that outputs the members of that list; formally, that the function $n \mapsto a_n$ is computable. And the sequence $n \mapsto p_n$ of prime numbers is perfectly computable, e.g. using the sieve of Eratosthenes.
I also do not agree that we don't know what the prime numbers are. Again, there's a rigorous formal definition of what it means to know a set $S \subseteq \mathbb{N}$ of numbers, and it means you can write down a computer program that outputs "yes" if some number belongs to that set and "no" if it doesn't; formally, that the set is computable. And the set of prime numbers is again computable (note that this is a different statement from the above statement); this is exactly the statement that there exist computable primality tests, and e.g. trial division qualifies.
In both of theses cases instead of "computable" you could ask for "efficiently computable" but primality testing is pretty efficient both in practice and in theory.
Overall I would argue that in most senses that we "know" what the natural numbers are, it's also true that we "know" what the prime numbers are.
An example of a set of numbers I would argue we genuinely don't know is the set of Fermat primes; as far as I know we don't have an efficient algorithm for testing whether a Fermat number $F_n = 2^{2^n} + 1$ is prime, basically because they grow too fast, e.g. according to Wikipedia it seems we don't know whether $F_{33}$ is prime. We also don't know whether the set is finite or infinite.