What is the fastest known algorithm that generates all distinct prime numbers less than n?
Is it faster than Sieve of Atkin?
What is the fastest known algorithm that generates all prime numbers?
I assume you mean: Given $n$, what is the fastest known algorithm that generates all prime numbers $p \le n$ ? Currently it is the Sieve of Atkin.
And what is the fastest known algorithm that generates any infinite subset of the prime numbers?
Again, I assume you mean: Given $n$, how fast can I generate $n$ distinct primes? There might be a faster method than the Sieve of Atkin, but I don't know of any. A good question!
And is there a theoretical lowest possible O(n) of such programs?
Is $n$ the number of primes you want to generate? Then it would take $O(n)$ operations just to store them in memory. So yes. But if you want to generate all primes $p \le n$ , the Sieve of Atkin has time complexity $O(n/\log \log n)$ . So no.
I recently just chanced upon a particular logic. All prime numbers either fall in multiples of 6n+1 or 6n-1 except for 2 and 3.
Using this the search space could be considerably reduced for applying any sieve such as Atkins or Eratosthenes. Hence making it a slightly better form of finding primes.
The sieve of Eratosthenes is one of the most efficient ways to find all the prime number less than n.
Odd numbers can be wheeled in multiplications to output only odd composite numbers. Then the odd numbers that are not output are the prime numbers.
Now each inner loop can stop at a multiplication that reaches the value of an upper-bound and the outer loop can stop at the square root of the upper-bound.
Furthermore the loops from the number 11 can increment with 2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,8,6,4,6,2,4,6,2,6,6,4,2,4,6,2,6,4,2,4,2,10,2,10 ... as continuting and repeating. That's a big performance gain because multiples of 3, 5, and 7 are removed from the sequence of odd numbers.
A prime number application really works best when outputting prime numbers between an upper bound and the upper bound - n. Then the application appears to be just scrolling sections (or jumping to sections) of a list on each computation. And in this case the loop increments are really only needed on the outer loop because a single division operation jumps over unneeded sections of the inner loop. Of course, array subscript locations can work with translations such that the same array can handle any of the segmented computations and then not use very much memory.