Prime Numbers

3.2 Sieving 121
3.2 Sieving
Sieving can be a highly efficient means of determining primality and factoring
when one is interested in the results for every number in a large, regularly
spaced set of integers. On average, the number of arithmetic operations spent
per number in the set can be very small, essentially bounded.
3.2.1 Sieving to recognize primes
Most readers are likely to be familiar with the sieve of Eratosthenes. In its
most common form it is a device for finding the primes up to some number
N. StartwithanarrayofN− 1 “ones,” corresponding to the numbers from
2toN. The first one corresponds to “2,” so the ones in locations 4, 6, 8,
and so on, are all changed to zeros. The next one is in the position “3,” and
we read this as an instruction to change any ones in locations 6, 9, 12, and
so on, into zeros. (Entries that are already zeros in these locations are left
unchanged.) We continue in this fashion. If the next entry one corresponds
to “p,” we change to zero any entry one at locations 2p, 3p, 4p, and so on.
However, if p is so large that p2 >N, we may stop this process. This exit
point can be readily detected by noticing that when we attempt to sieve by p
there are no changes of ones to zeros to be made. At this point the one entries
in the list correspond to the primes not exceeding N, while the zero entries
correspond to the composites.
In passing through the list 2p, 3p, 4p, and so on, one starts from the initial
number p and sequentially adds p until we arrive at a number exceeding N.
Thus the arithmetic operations in the sieve are all additions. The number of
steps in the sieve of Eratosthenes is proportional to
N/p, wherep runs
over primes. But
p≤N
N
p
p≤N
= N ln ln N + O(N); (3.1)
see Theorem 427 in [Hardy and Wright 1979]. Thus, the number of steps
needed per number up to N is proportional to ln ln N. It should be noted
that ln ln N, though it does go to infinity, does so very slowly. For example,
ln ln N