I have a little tool called Prime Grid Explorer at https://susam.net/primegrid.html that I wrote for my own amusement. It can display all primes below 3317044064679887385961981 (an 82-bit integer).

The largest three primes it can show are

  3317044064679887385961783
  3317044064679887385961801
  3317044064679887385961813

So essentially it can test all 81-bit integers and some 82-bit integers for primality. It does so using the Miller-Rabin primality test with prime bases derived from https://oeis.org/A014233 (OEIS A014233). The algorithm is implemented in about 80 lines of plain JavaScript. If you view the source, look for the function isPrimeByMR.

The Miller-Rabin test is inherently probabilistic. It tests whether a number is a probable prime by checking whether certain number theoretic congruence relations hold for a given base a. The test can yield false positives, that is, a composite number may pass the test. But it cannot have false negatives, so a number that fails the test is definitely composite. The more bases for which the test holds, the more likely it is that the tested number is prime. It has been computationally verified that there are no false positives below 3317044064679887385961981 when tested with prime bases 2, 3, 5, ..., 41. So although the algorithm is probabilistic, it functions as a deterministic test for all numbers below this bound when tested with these 13 bases.

There is also the segmented Sieve of Eratosthenes. It has a simlar performance but uses much less memory: the number of prime numbers from 2 to sqrt(n). For example, for n = 1000000, the RAM has to store only 168 additional numbers.

I use this algorithm here https://surenenfiajyan.github.io/prime-explorer/

You can combine the Sieve and Wheel techniques to reduce the memory requirements dramatically. There's no need to use a bit for numbers that you already know can't be prime. You can find a Python implementation at https://stackoverflow.com/a/62919243/5987

If you take all 53 8 bit primes, you can use modular arithmetic with a residue base to work with numbers up to

64266330917908644872330635228106713310880186591609208114244758680898150367880703152525200743234420230

This would require 334 bits.

Do you know the https://en.wikipedia.org/wiki/Sieve_of_Atkin? It's mind-blowing.

This got me through many of the first 100 problems on Project Euler:

    n = 1000000 # must be even
    sieve = [True] * (n/2)
    for i in range(3,int(n**0.5)+1,2):
        if sieve[i/2]: sieve[i*i/2::i] = [False] * ((n-i*i-1)/(2*i)+1)
    …
    # x is prime if x%2 and sieve[x/2]

I always like seeing implementations that start from trial division and gradually introduce optimizations like wheel factorization.

It makes the trade-offs much clearer than jumping straight to a complex sieve.

Very well written

> There is a long way to go from here. Kim Walisch's primesieve can generate all 32-bit primes in 0.061s (though this is without writing them to a file)

Oh, come on, just use a bash indirection and be done with it. It takes 1 minute and you had another result for comparison

Why include writing the primes to a file instead of, say, standard output? That increases the optimization space drastically and the IO will eclipse all the careful bitwise math

Does having the primes in a file even allow faster is-prime lookup of a number?

there are also very fast primality tests that work statistically. It's called Miller-Rabin, I tested in the browser here[1] and it can do them all in about three minutes on my phone.

[1] https://claude.ai/public/artifacts/baa198ed-5a17-4d04-8cef-7...