Pair of prime numbers with a given sum and minimum absolute difference

Given an integer ‘sum’ (less than 10^8), the task is to find a pair of prime numbers whose sum is equal to the given ‘sum’ 
Out of all the possible pairs, the absolute difference between the chosen pair must be minimum. 
If the ‘sum’ cannot be represented as a sum of two prime numbers then print “Cannot be represented as sum of two primes”.
Examples:

Input : Sum = 1002
Output : Primes: 499 503
Explanation
1002 can be represented as sum of many prime number pairs
such as
499 503
479 523
461 541
439 563
433 569
431 571
409 593
401 601...
But 499 and 503 is the only pair which has minimum difference 

Input :Sum = 2002
Output : Primes: 983 1019

Solution

• We will create a sieve of Eratosthenes which will store all the prime numbers and check whether a number is prime or not in O(1) time.
• Now, to find two prime numbers with sum equal to the given variable, ‘sum’. We will start a loop from sum/2 to 1 (to minimize the absolute difference) and check whether the loop counter ‘i’ and ‘sum-i’ are both prime.
• If they are prime then we will print them and break out of the loop.
• If the ‘sum’ cannot be represented as a sum of two prime numbers then we will print “Cannot be represented as sum of two primes”.

Below is the implementation of the above solution: 

C++

// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
#define MAX 100000000

// stores whether a number is prime or not
bool prime[MAX + 1];

// create the sieve of eratosthenes
void SieveOfEratosthenes()
{
    // Create a boolean array "prime[0..n]" and initialize
    // all entries it as true. A value in prime[i] will
    // finally be false if i is Not a prime, else true.
    memset(prime, true, sizeof(prime));

    prime[1] = false;

    for (int p = 2; p * p <= MAX; p++) {

        // If prime[p] is not changed, then it is a prime
        if (prime[p] == true) {

            // Update all multiples of p as non-prime
            for (int i = p * 2; i <= MAX; i += p)
                prime[i] = false;
        }
    }
}

// find the two prime numbers with minimum
// difference and whose sum is equal to
// variable sum
void find_Prime(int sum)
{

    // start from sum/2 such that
    // difference between i and sum-i will be
    // minimum
    for (int i = sum / 2; i > 1; i--) {

        // if both 'i' and 'sum - i' are prime then print
        // them and break the loop
        if (prime[i] && prime[sum - i]) {
            cout << i << " " << (sum - i) << endl;
            return;
        }
    }
    // if there is no prime
    cout << "Cannot be represented as sum of two primes" << endl;
}

// Driver code
int main()
{
    // create the sieve
    SieveOfEratosthenes();

    int sum = 1002;

    // find the primes
    find_Prime(sum);

    return 0;
}

Java

//Java implementation of the above approach

class GFG {

    static final int MAX = 100000000;

    // stores whether a number is prime or not
    static boolean prime[] = new boolean[MAX + 1];

    // create the sieve of eratosthenes
    static void SieveOfEratosthenes() {
        // Create a boolean array "prime[0..n]" and initialize
        // all entries it as true. A value in prime[i] will
        // finally be false if i is Not a prime, else true.
        for (int i = 0; i < prime.length; i++) {
            prime[i] = true;
        }
        prime[1] = false;

        for (int p = 2; p * p <= MAX; p++) {

            // If prime[p] is not changed, then it is a prime
            if (prime[p] == true) {

                // Update all multiples of p as non-prime
                for (int i = p * 2; i <= MAX; i += p) {
                    prime[i] = false;
                }
            }
        }
    }

    // find the two prime numbers with minimum
    // difference and whose sum is equal to
    // variable sum
    static void find_Prime(int sum) {

        // start from sum/2 such that
        // difference between i and sum-i will be
        // minimum
        for (int i = sum / 2; i > 1; i--) {

            // if both 'i' and 'sum - i' are prime then print
            // them and break the loop
            if (prime[i] && prime[sum - i]) {
                System.out.println(i + " " + (sum - i));
                return;
            }
        }
        // if there is no prime
        System.out.println("Cannot be represented as sum of two primes");
    }
    public static void main(String []args) {
        // create the sieve
        SieveOfEratosthenes();
        int sum = 1002;
        // find the primes
        find_Prime(sum);
    }
}
/*This code is contributed by 29AjayKumar*/

Python3

# Python 3 implementation of the above approach
from math import sqrt

# stores whether a number is prime or not

# create the sieve of eratosthenes
def SieveOfEratosthenes():
    MAX = 1000001

    # Create a boolean array "prime[0..n]" and
    # initialize all entries it as true. A value
    # in prime[i] will finally be false if i is
    # Not a prime, else true.
    prime = [True for i in range(MAX + 1)]

    prime[1] = False

    for p in range(2, int(sqrt(MAX)) + 1, 1):

        # If prime[p] is not changed,
        # then it is a prime
        if (prime[p] == True):

            # Update all multiples of p
            # as non-prime
            for i in range(p * 2, MAX + 1, p):
                prime[i] = False

    return prime

# find the two prime numbers with minimum
# difference and whose sum is equal to
# variable sum
def find_Prime(sum):

    # start from sum/2 such that difference
    # between i and sum-i will be minimum
    # create the sieve
    prime = SieveOfEratosthenes()
    i = int(sum / 2)
    while(i > 1):

        # if both 'i' and 'sum - i' are prime
        # then print them and break the loop
        if (prime[i] and prime[sum - i]):
            print(i, (sum - i))
            return

        i -= 1

    # if there is no prime
    print("Cannot be represented as sum",
                         "of two primes")

# Driver code
if __name__ == '__main__':

    sum = 1002

    # find the primes
    find_Prime(sum)

# This code is contributed by
# Shashank_Sharma

C#

// C# implementation of the
// above approach
class GFG
{

static int MAX = 1000000;

// stores whether a number is
// prime or not
static bool[] prime = new bool[MAX + 1];

// create the sieve of eratosthenes
static void SieveOfEratosthenes()
{
    // Create a boolean array "prime[0..n]"
    // and initialize all entries it as true.
    // A value in prime[i] will finally be
    // false if i is Not a prime, else true.
    for (int i = 0; i < prime.Length; i++)
    {
        prime[i] = true;
    }
    prime[1] = false;

    for (int p = 2; p * p <= MAX; p++)
    {

        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p] == true)
        {

            // Update all multiples of p
            // as non-prime
            for (int i = p * 2;
                     i <= MAX; i += p)
            {
                prime[i] = false;
            }
        }
    }
}

// find the two prime numbers with
// minimum difference and whose sum
// is equal to variable sum
static void find_Prime(int sum)
{

    // start from sum/2 such that
    // difference between i and sum-i
    // will be minimum
    for (int i = sum / 2; i > 1; i--)
    {

        // if both 'i' and 'sum - i'
        // are prime then print
        // them and break the loop
        if (prime[i] && prime[sum - i])
        {
            System.Console.WriteLine(i + " " +
                                    (sum - i));
            return;
        }
    }

    // if there is no prime
    System.Console.WriteLine("Cannot be represented " +
                               "as sum of two primes");
}

// Driver Code
static void Main()
{
    // create the sieve
    SieveOfEratosthenes();
    int sum = 1002;

    // find the primes
    find_Prime(sum);
}
}

// This code is contributed by mits

Javascript

<script>

// Javascript implementation of the above approach
var MAX = 1000001;

// stores whether a number is prime or not
var prime = Array(MAX+1).fill(true);

// create the sieve of eratosthenes
function SieveOfEratosthenes()
{
    // Create a boolean array "prime[0..n]" and initialize
    // all entries it as true. A value in prime[i] will
    // finally be false if i is Not a prime, else true.
    prime[1] = false;

    for (var p = 2; p * p <= MAX; p++) {

        // If prime[p] is not changed, then it is a prime
        if (prime[p] == true) {

            // Update all multiples of p as non-prime
            for (var i = p * 2; i <= MAX; i += p)
                prime[i] = false;
        }
    }
}

// find the two prime numbers with minimum
// difference and whose sum is equal to
// variable sum
function find_Prime(sum)
{

    // start from sum/2 such that
    // difference between i and sum-i will be
    // minimum
    for (var i = parseInt(sum / 2); i > 1; i--) {

        // if both 'i' and 'sum - i' are prime then print
        // them and break the loop
        if (prime[i] && prime[sum - i]) {
            document.write( i + " " + (sum - i) + "<br>");
            return;
        }
    }
    // if there is no prime
    document.write( "Cannot be represented as sum of two primes" + "<br>");
}

// Driver code
// create the sieve
SieveOfEratosthenes();
var sum = 1002;
// find the primes
find_Prime(sum);


</script>

Output:

499 503

Time Complexity: O(n + MAX^3/2)

Auxiliary Space: O(MAX)