one of the key goals of number theory is to understand prime numbers, in particular, to predict which numbers are prime and which are not. If we would know the exact function $\pi(x)$ which denotes how many prime numbers there are smaller than $x$, then we could ask whether $x$ is prime by computing $\pi(x)-\pi(x-1)$.
But I have heard that we don't really understand the distribution of prime numbers $\pi(x)$. we have the prime number theorem to approximate $\pi(x)$, and we can use the Riemann hypothesis (if it is true) to improve upon that approximation.
But it is not exactly clear to me when mathematicians will be satisfied and say: "yes, now we can predict which numbers are prime", or "now we understand the distribution $\pi(x)$".
Obviously, we do not mean that we can compute $\pi(x)$. In that case, we can, it is written as follows:
bool isPrime(x){
for {y=1;y<x;y++}
{
if isDivisible(x,y) then return false;
}
return true;
}
When are mathematicians satisfied?
When $\pi(x)$ can be computed efficiently?
When we "understand" the distribution $\pi(x)$? (Under what conditions would or would we not "understand" it?)
The point is : It can be determined, even in a very easy way , whether a given positive integer $N$ is a prime number or not. We say : This problem is decideable.
For practical purposes, we do not only want to know the existence of such an algorithm, but we want to find an efficient algorithm ("efficient" usually means with a time growing polynomial with the input). In fact, primality checking was recently be proven to be in $P$
But still , to test a huge number , lets say, $1$ million digits , is extremely time consuming. We do not have an algorithm telling us "immediately" whether the number is prime. Such an "immediate" result is possible , if we for example calculate $\gcd(a,b)$ with huge integers $a$ and $b$.
In this sense, primes are "not predictable", and $\pi(x)$ can neither be determined exactly enough to find huge primes.