# 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 = 1002Output :Primes: 499 503Explanation1002 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 differenceInput :Sum = 2002Output :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; ` `} ` |

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## 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*/` |

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## 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 ` |

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**Output:**

499 503

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