public class Goldbach  {

    private static int j;
    private static int k;
    private static int [] primes;
    private static int [] just_primes;
    private static int [] conjunct;

    public static void main(String[] args)
    {

        primes = new int[2000000];
        for ( int i=0; i<2000000; i++ ) {
            primes[i] = i+1;
        }

        for ( int i=1; i<1000000; i++ ) {
            if ( primes[i] != 0 ) {
                j = primes[i];
                k = j;
                while ( k <= 2000000 ) {
                    k += j;
                    if ( k <= 2000000 ) {
                        primes[ k-1 ] = 0;
                    }
                }
            }
        }

        just_primes = new int [primes.length];
        for ( int i = 0; i< primes.length-1 ; i++ ) {
            just_primes[i] = 0;
        }

        k = 0;
        for ( int i = 1; i< primes.length-1 ; i++ ) {
            if ( primes[i] > 0 ) {
                just_primes[k] = primes[i];
                k++;
            }
        }

        conjunct = new int[1000001];
        for ( int i = 0; i<1000001 ; i++ ) {
            conjunct[i] = 0;
        }

        int p;
        for ( int i = 0; i< k ; i++ ) {
            for ( int j = i; j< k ; j++ ) {
                int w = just_primes[i]+just_primes[j];
                if ( w >= 1000000 && w <= 2000000 ) {
                    p = w - 1000000;
                    conjunct[p]++;
                }
            }
        }

        int big1 = 0;
        int big2 = 0;
        int big3 = 0;
        int n1 = 0;
        int n2 = 0;
        int n3 = 0;

        for ( int i = 0; i<1000001 ; i++ ) {
            if ( conjunct[i] >= big1 ) {
                big3 = big2;
                big2 = big1;
                big1 = conjunct[i];
                n3 = n2;
                n2 = n1;
                n1 = i;
            } else {
                if ( conjunct[i] >= big2 ) {
                    big3 = big2;
                    big2 = conjunct[i];
                    n3 = n2;
                    n2 = i;
                } else {
                    if ( conjunct[i] >= big3 ) {
                        big3 = conjunct[i];
                        n3 = i;
                    }
                }
            }
        }


        System.out.println( "Largest conjunct 1: " + n1 + " " + 1000000+big1 + " pairs" );
        System.out.println( "Largest conjunct 2: " + n2 + " " + 1000000+big2 + " pairs" );
        System.out.println( "Largest conjunct 3: " + n3 + " " + 1000000+big3 + " pairs" );

    }

}

Largest conjunct 1: 981980 100000027988 pairs
Largest conjunct 2: 951950 100000027802 pairs
Largest conjunct 3: 995630 100000027730 pairs