Prime Numbers

126 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
It is important to be able to sieve the consecutive values of a polynomial
for B-smooth numbers, as in Section 3.2.5. All of the ideas of that section
port most naturally to the ideas of this section.
3.2.7 Theoretical considerations
The complexity N ln ln N of the sieve of Eratosthenes may be reduced
somewhat by several clever arguments. The following algorithm is based
on ideas of Mairson and Pritchard (see [Pritchard 1981]). It requires only
O(N/ ln ln N) steps, where each step is either for bookkeeping or an addition
with integers at most N. (Note that an explicit pseudocode display for a
rudimentary Eratosthenes sieve appears in Section 3.2.2.)
Algorithm 3.2.2 (Fancy Eratosthenes sieve). We are given a number N ≥
4. This algorithm finds the set of primes in [1,N]. Letpl denote the l-th prime,
let Ml = p1p2 ···pl, and let Sl denote the set of numbers in [1,N] that are
coprime to Ml. Note that if pm+1 > √ N, then the set of primes in [1,N] is
(Sm \{1}) ∪{p1,p2,...,pm}. The algorithm recursively finds Sk,Sk+1,...,Sm
starting from a moderately sized initial value k and ending with m = π( √ N).
1. [Setup]
Set k as the integer with Mk ≤ N/ ln N