Example 1-15 computes the largest prime number less than a specified value, using the Sieve of Eratosthenes algorithm. The algorithm finds primes by eliminating multiples of all lower prime numbers. Like Example 1-14, this example introduces no new Java syntax, but is a nice, nontrivial program with which to end this chapter. The program may seem deceptively simple, but there's actually a fair bit going on, so be sure you understand how it is ruling out prime numbers.
package je3.basics;
/**
* This program computes prime numbers using the Sieve of Eratosthenes
* algorithm: rule out multiples of all lower prime numbers, and anything
* remaining is a prime. It prints out the largest prime number less than
* or equal to the supplied command-line argument.
**/
public class Sieve {
public static void main(String[ ] args) {
// We will compute all primes less than the value specified on the
// command line, or, if no argument, all primes less than 100.
int max = 100; // Assign a default value
try { max = Integer.parseInt(args[0]); } // Parse user-supplied arg
catch (Exception e) { } // Silently ignore exceptions.
// Create an array that specifies whether each number is prime or not.
boolean[ ] isprime = new boolean[max+1];
// Assume that all numbers are primes, until proven otherwise.
for(int i = 0; i <= max; i++) isprime[i] = true;
// However, we know that 0 and 1 are not primes. Make a note of it.
isprime[0] = isprime[1] = false;
// To compute all primes less than max, we need to rule out
// multiples of all integers less than the square root of max.
int n = (int) Math.ceil(Math.sqrt(max)); // See java.lang.Math class
// Now, for each integer i from 0 to n:
// If i is a prime, then none of its multiples are primes,
// so indicate this in the array. If i is not a prime, then
// its multiples have already been ruled out by one of the
// prime factors of i, so we can skip this case.
for(int i = 0; i <= n; i++) {
if (isprime[i]) // If i is a prime,
for(int j = 2*i; j <= max; j = j + i) // loop through multiples
isprime[j] = false; // they are not prime.
}
// Now go look for the largest prime:
int largest;
for(largest = max; !isprime[largest]; largest--) ; // empty loop body
// Output the result
System.out.println("The largest prime less than or equal to " + max +
" is " + largest);
}
}