Dietzfelbinger M. Primality testing in polynomial time ... - tiera.ru

40 3. Fundamentals from Number Theory
It is not very hard to see that the numbers that are left unmarked are
exactly the prime numbers in the interval [2,n]. Indeed, no prime number is
ever marked, since only numbers sj with s ≥ j ≥ 2 are marked. Conversely, if
k ≤ n is composite, then write k as a product ab with b ≥ a ≥ 2. Obviously,
then, a ≤ √ k ≤ √ n. By Lemma 3.5.2 there is some prime number p that
divides a. Thenwemaywritek = sp for some s ≥ b, hence s ≥ p. Whenj
attains the value p, this prime number turns out to be unmarked, and the
number k becomes marked as the multiple sp of p.
A slightly more elaborate version of the Sieve of Eratosthenes, given next,
even marks each composite number in [2,n] with its smallest prime divisor.
Algorithm 3.5.4 (The Sieve of Eratosthenes)
Input: Integer n ≥ 2
Method:
1 m[2..n]: array of integer;
2 for j from 2 to n do m[j] ← 0;
3 j ← 2;
4 while j · j ≤ n do
5 if m[j] =0then
6 i ← j · j;
7 while i ≤ n do
8 if m[i] =0then m[i] ← j;
9 i ← i + j;
10 j ← j +1;
11 return m[2..n];
The algorithm varies the idea sketched above as follows. Instead of attaching
a “mark” to k, m[k] is assigned a nonzero value. The j-loop (lines
4–10) treatsthenumbersj =2, 3,...,⌊ √ n⌋, inorder.Ifm[j] turns out to
be nonzero when j has value j, theninthei-loop (lines 7–9) theunmarked
multiples of j are marked with j. — The algorithm is easily analyzed. As
before, if k ≤ n is a prime number, then the value m[k] stays 0 throughout.
Assume now that k ≤ n is composite. Consider the smallest prime number
p that is a divisor of k, andwritek = sp for some s ≥ 2. By Lemma 3.5.2,
s is divisible by some prime number p ′ , which then must exceed p; hence
p 2 ≤ sp = k ≤ n. From the algorithm it is then obvious that in the iteration
of the j-loop (lines 4–10) in which the variable j contains p the array component
m[k] will be assigned the value p. (It cannot get a value different from 0
before that, since this would imply that k is divisible by some number j