Binomial with factorization

import java.math.BigInteger;
import java.util.ArrayList;

public class Binomial {
    //PRIME SIEVE FOR DECOMPOSITION
    static boolean[] bs;
    static ArrayList<Integer> all_primes;
    static long size;
    public static void sieve(long upperbound){
        size = upperbound + 1; // we add 1 to include upperbound
        bs = new boolean[(int) size];
        all_primes = new ArrayList<>();

        for(long i = 2 ; i < size ; i++){
            if(!bs[(int)i]){
                for(long j = i*i ; j < size ; j+=i){
                    bs[(int)j] = true;
                }
                all_primes.add((int)i);
            }
        }
    }
    //PRIME SIEVE CODE ENDS

    //Implementation with prime factor decomposition. VERY FAST for big values of N.
    //(Source: Computing Binomial Coefficients, P. Goetgheluck)
    //Requires a list of all primes less than or equal to N (generated once by the sieve)
    //REMEMBER TO CALL SIEVE FUNCTION BEFORE CALLING COMBINATION, ALSO MAKE SURE ALL PRIMES LESS THAN OR EQUAL TO N ARE GENERATED
    public static BigInteger combination(int N, int K){
        if ((K == 0) || (K == N)) return BigInteger.ONE;
        K = Math.min(N-K, K);
        int nk = N - K;
        int rootN = (int) Math.sqrt(N);
        int n_half = N/2;
        ArrayList<Integer> exponents = new ArrayList<>(); //list of the exponents of the prime factorization.
        //example: exponents = {2, 0, 1} means that the answer is 2^2 * 3^0 * 5^1

        for(int prime : all_primes){
            if(prime > N) break;

            if (prime > nk) exponents.add(1);
            else if (prime > n_half) exponents.add(0);
            else if (prime > rootN){
                if (N % prime < K % prime) exponents.add(1);
                else exponents.add(0);
            }
            else{
                int r = 0, tmpN = N, tmpK = K, a, b, total = 0;

                while(tmpN > 0){
                    a = tmpN % prime;
                    b = (tmpK%prime) + r;
                    tmpN /= prime;
                    tmpK /= prime;

                    if(a<b){
                        total++;
                        r = 1;
                    }
                    else r = 0;
                }
                exponents.add(total);
            }
        }

        //joining all prime factors in one BigInteger
        BigInteger R = BigInteger.ONE;
        for(int i = 0, len = exponents.size(); i<len; i++){
            int exp = exponents.get(i);
            if(exp > 0) R = R.multiply(BigInteger.valueOf((long) Math.pow(all_primes.get(i), exp)));
        }

        return R;
    }

    public static void main(String[] args) {
        int N = 3;
        sieve(N);
        System.out.println(combination(N, 2));
    }
}