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Project Euler #46: Goldbach's other conjecture

Question
It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
9 = 7 + 2×12
15 = 7 + 2×22
21 = 3 + 2×32
25 = 7 + 2×32
27 = 19 + 2×22
33 = 31 + 2×12
It turns out that the conjecture was false.
What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

Answer : 5777

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Solution


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import java.util.*;

public class Solution {
    private static boolean[] prime = null;

    public static void main(String[] args) {
        /* Enter your code here. Read input from STDIN. Print output to STDOUT. Your class should be named Solution. */
        Scanner in = new Scanner(System.in);
        int numberOfTestCases = in.nextInt();
        generatePrimes();

        for(int i=0;i<numberOfTestCases;i++)
        {
            int res=0;
            int n = in.nextInt();
            for(int j=2;j<=n;j++)
            {
                if(prime[j])
                {
                    int diff=n-j;
                    if(diff%2==0 && isPerfectSquare(diff/2))
                        res+=1;
                }
            }
            System.out.println(res);
        }
    }

    public static boolean isPerfectSquare(int input)
    {
        int root = (int) Math.sqrt(input);
        return input == root * root;
    }

    public static void generatePrimes()
    {
        int limit = 1000000;
        prime = new boolean[limit+1];
        prime[2] = true;
        prime[3] = true;
        int root = (int) Math.ceil(Math.sqrt(limit));

        //Sieve of Atkin for prime number generation
        for (int x = 1; x < root; x++)
        {
            for (int y = 1; y < root; y++)
            {
                int n = 4 * x * x + y * y;
                if (n <= limit && (n % 12 == 1 || n % 12 == 5))
                    prime[n] = !prime[n];

                n = 3 * x * x + y * y;
                if (n <= limit && n % 12 == 7)
                    prime[n] = !prime[n];

                n = 3 * x * x - y * y;
                if ((x > y) && (n <= limit) && (n % 12 == 11))
                    prime[n] = !prime[n];
            }
        }

        for (int i = 5; i <= root; i++)
        {
            if (prime[i])
            {
                for (int j = i * i; j < limit; j += i * i)
                {
                    prime[j] = false;
                }
            }
        }
    }
}
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