How to create the most compact mapping n → isprime(n) up to a limit N?

The fastest algorithm for general prime testing is AKS. The Wikipedia article describes it at lengths and links to the original paper.

If you want to find big numbers, look into primes that have special forms like Mersenne primes.

The algorithm I usually implement (easy to understand and code) is as follows (in Python):

def isprime(n):
    """Returns True if n is prime."""
    if n == 2:
        return True
    if n == 3:
        return True
    if n % 2 == 0:
        return False
    if n % 3 == 0:
        return False

    i = 5
    w = 2

    while i * i <= n:
        if n % i == 0:
            return False

        i += w
        w = 6 - w

    return True

It's a variant of the classic O(sqrt(N)) algorithm. It uses the fact that a prime (except 2 and 3) is of form 6k - 1 or 6k + 1 and looks only at divisors of this form.

Sometimes, If I really want speed and the range is limited, I implement a pseudo-prime test based on Fermat's little theorem. If I really want more speed (i.e. avoid O(sqrt(N)) algorithm altogether), I precompute the false positives (see Carmichael numbers) and do a binary search. This is by far the fastest test I've ever implemented, the only drawback is that the range is limited.

There are many ways to do the primality test.

There isn't really a data structure for you to query. If you have lots of numbers to test, you should probably run a probabilistic test since those are faster, and then follow it up with a deterministic test to make sure the number is prime.

You should know that the math behind the fastest algorithms is not for the faint of heart.

The best method, in my opinion, is to use what's gone before.

There are lists of the first N primes on the internet with N stretching up to at least fifty million. Download the files and use them, it's likely to be much faster than any other method you'll come up with.

If you want an actual algorithm for making your own primes, Wikipedia has all sorts of good stuff on primes here, including links to the various methods for doing it, and prime testing here, both probability-based and fast-deterministic methods.

There should be a concerted effort to find the first billion (or even more) primes and get them published on the net somewhere so people can stop doing this same job over and over and over and ... :-)