Program for Goldbach’s Conjecture (Two Primes with given Sum)

Goldbach’s conjecture is one of the oldest and best-known unsolved problems in number theory of mathematics. Every even integer greater than 2 can be expressed as the sum of two primes.

Examples:

Input :  n = 44
Output :   3 + 41 (both are primes)

Input :  n = 56
Output :  3 + 53  (both are primes)

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Find the prime numbers using Sieve of Sundaram
Check if entered number is an even number greater than 2 or not, if no return.
If yes, then one by one subtract a prime from N and then check if the difference is also a prime, if yes then express it as a sum.

C++

// C++ program to implement Goldbach's conjecture 
#include<bits/stdc++.h> 
using namespace std; 
const int MAX = 10000; 
  
// Array to store all prime less than and equal to 10^6 
vector <int> primes; 
  
// Utility function for Sieve of Sundaram 
void sieveSundaram() 
{ 
    // In general Sieve of Sundaram, produces primes smaller 
    // than (2*x + 2) for a number given number x. Since 
    // we want primes smaller than MAX, we reduce MAX to half 
    // This array is used to separate numbers of the form 
    // i + j + 2*i*j from others where 1 <= i <= j 
    bool marked[MAX/2 + 100] = {0}; 
  
    // Main logic of Sundaram. Mark all numbers which 
    // do not generate prime number by doing 2*i+1 
    for (int i=1; i<=(sqrt(MAX)-1)/2; i++) 
        for (int j=(i*(i+1))<<1; j<=MAX/2; j=j+2*i+1) 
            marked[j] = true; 
  
    // Since 2 is a prime number 
    primes.push_back(2); 
  
    // Print other primes. Remaining primes are of the 
    // form 2*i + 1 such that marked[i] is false. 
    for (int i=1; i<=MAX/2; i++) 
        if (marked[i] == false) 
            primes.push_back(2*i + 1); 
} 
  
// Function to perform Goldbach's conjecture 
void findPrimes(int n) 
{ 
    // Return if number is not even or less than 3 
    if (n<=2 || n%2 != 0) 
    { 
        cout << "Invalid Input \n"; 
        return; 
    } 
  
    // Check only upto half of number 
    for (int i=0 ; primes[i] <= n/2; i++) 
    { 
        // find difference by subtracting current prime from n 
        int diff = n - primes[i]; 
  
        // Search if the difference is also a prime number 
        if (binary_search(primes.begin(), primes.end(), diff)) 
        { 
            // Express as a sum of primes 
            cout << primes[i] << " + " << diff << " = "
                 << n << endl; 
            return; 
        } 
    } 
} 
  
// Driver code 
int main() 
{ 
    // Finding all prime numbers before limit 
    sieveSundaram(); 
  
    // Express number as a sum of two primes 
    findPrimes(4); 
    findPrimes(38); 
    findPrimes(100); 
  
    return 0; 
} 

Java

// Java program to implement Goldbach's conjecture 
import java.util.*; 
  
class GFG 
{ 
      
static int MAX = 10000; 
  
// Array to store all prime less  
// than and equal to 10^6 
static ArrayList<Integer> primes = new ArrayList<Integer>(); 
  
// Utility function for Sieve of Sundaram 
static void sieveSundaram() 
{ 
    // In general Sieve of Sundaram, produces  
    // primes smaller than (2*x + 2) for  
    // a number given number x. Since 
    // we want primes smaller than MAX, 
    // we reduce MAX to half This array is  
    // used to separate numbers of the form 
    // i + j + 2*i*j from others where 1 <= i <= j 
    boolean[] marked = new boolean[MAX / 2 + 100]; 
  
    // Main logic of Sundaram. Mark all numbers which 
    // do not generate prime number by doing 2*i+1 
    for (int i = 1; i <= (Math.sqrt(MAX) - 1) / 2; i++) 
        for (int j = (i * (i + 1)) << 1; j <= MAX / 2; j = j + 2 * i + 1) 
            marked[j] = true; 
  
    // Since 2 is a prime number 
    primes.add(2); 
  
    // Print other primes. Remaining primes are of the 
    // form 2*i + 1 such that marked[i] is false. 
    for (int i = 1; i <= MAX / 2; i++) 
        if (marked[i] == false) 
            primes.add(2 * i + 1); 
} 
  
// Function to perform Goldbach's conjecture 
static void findPrimes(int n) 
{ 
    // Return if number is not even or less than 3 
    if (n <= 2 || n % 2 != 0) 
    { 
        System.out.println("Invalid Input "); 
        return; 
    } 
  
    // Check only upto half of number 
    for (int i = 0 ; primes.get(i) <= n / 2; i++) 
    { 
        // find difference by subtracting  
        // current prime from n 
        int diff = n - primes.get(i); 
  
        // Search if the difference is  
        // also a prime number 
        if (primes.contains(diff)) 
        { 
            // Express as a sum of primes 
            System.out.println(primes.get(i) +  
                        " + " + diff + " = " + n); 
            return; 
        } 
    } 
} 
  
// Driver code 
public static void main (String[] args)  
{ 
    // Finding all prime numbers before limit 
    sieveSundaram(); 
  
    // Express number as a sum of two primes 
    findPrimes(4); 
    findPrimes(38); 
    findPrimes(100); 
} 
} 
  
// This code is contributed by mits 

Python3

# Python3 program to implement Goldbach's  
# conjecture 
import math 
MAX = 10000; 
  
# Array to store all prime less  
# than and equal to 10^6 
primes = []; 
  
# Utility function for Sieve of Sundaram 
def sieveSundaram(): 
      
    # In general Sieve of Sundaram, produces  
    # primes smaller than (2*x + 2) for a  
    # number given number x. Since we want 
    # primes smaller than MAX, we reduce  
    # MAX to half. This array is used to  
    # separate numbers of the form i + j + 2*i*j  
    # from others where 1 <= i <= j 
    marked = [False] * (int(MAX / 2) + 100); 
  
    # Main logic of Sundaram. Mark all  
    # numbers which do not generate prime 
    # number by doing 2*i+1 
    for i in range(1, int((math.sqrt(MAX) - 1) / 2) + 1): 
        for j in range((i * (i + 1)) << 1,  
                        int(MAX / 2) + 1, 2 * i + 1): 
            marked[j] = True; 
  
    # Since 2 is a prime number 
    primes.append(2); 
  
    # Print other primes. Remaining primes  
    # are of the form 2*i + 1 such that  
    # marked[i] is false. 
    for i in range(1, int(MAX / 2) + 1): 
        if (marked[i] == False): 
            primes.append(2 * i + 1); 
  
# Function to perform Goldbach's conjecture 
def findPrimes(n): 
      
    # Return if number is not even  
    # or less than 3 
    if (n <= 2 or n % 2 != 0): 
        print("Invalid Input"); 
        return; 
  
    # Check only upto half of number 
    i = 0; 
    while (primes[i] <= n // 2): 
          
        # find difference by subtracting  
        # current prime from n 
        diff = n - primes[i]; 
  
        # Search if the difference is also 
        # a prime number 
        if diff in primes: 
              
            # Express as a sum of primes 
            print(primes[i], "+", diff, "=", n); 
            return; 
        i += 1; 
  
# Driver code 
  
# Finding all prime numbers before limit 
sieveSundaram(); 
  
# Express number as a sum of two primes 
findPrimes(4); 
findPrimes(38); 
findPrimes(100); 
  
# This code is contributed 
# by chandan_jnu 

C#

// C# program to implement Goldbach's conjecture 
using System; 
using System.Collections.Generic; 
  
class GFG 
{ 
      
static int MAX = 10000; 
  
// Array to store all prime less  
// than and equal to 10^6 
static List<int> primes = new List<int>(); 
  
// Utility function for Sieve of Sundaram 
static void sieveSundaram() 
{ 
    // In general Sieve of Sundaram, produces  
    // primes smaller than (2*x + 2) for  
    // a number given number x. Since 
    // we want primes smaller than MAX, 
    // we reduce MAX to half This array is  
    // used to separate numbers of the form 
    // i + j + 2*i*j from others where 1 <= i <= j 
    Boolean[] marked = new Boolean[MAX / 2 + 100]; 
  
    // Main logic of Sundaram. Mark all numbers which 
    // do not generate prime number by doing 2*i+1 
    for (int i = 1; i <= (Math.Sqrt(MAX) - 1) / 2; i++) 
        for (int j = (i * (i + 1)) << 1; j <= MAX / 2; j = j + 2 * i + 1) 
            marked[j] = true; 
  
    // Since 2 is a prime number 
    primes.Add(2); 
  
    // Print other primes. Remaining primes are of the 
    // form 2*i + 1 such that marked[i] is false. 
    for (int i = 1; i <= MAX / 2; i++) 
        if (marked[i] == false) 
            primes.Add(2 * i + 1); 
} 
  
// Function to perform Goldbach's conjecture 
static void findPrimes(int n) 
{ 
    // Return if number is not even or less than 3 
    if (n <= 2 || n % 2 != 0) 
    { 
        Console.WriteLine("Invalid Input "); 
        return; 
    } 
  
    // Check only upto half of number 
    for (int i = 0 ; primes[i] <= n / 2; i++) 
    { 
        // find difference by subtracting  
        // current prime from n 
        int diff = n - primes[i]; 
  
        // Search if the difference is  
        // also a prime number 
        if (primes.Contains(diff)) 
        { 
            // Express as a sum of primes 
            Console.WriteLine(primes[i] +  
                        " + " + diff + " = " + n); 
            return; 
        } 
    } 
} 
  
// Driver code 
public static void Main (String[] args)  
{ 
    // Finding all prime numbers before limit 
    sieveSundaram(); 
  
    // Express number as a sum of two primes 
    findPrimes(4); 
    findPrimes(38); 
    findPrimes(100); 
} 
} 
  
/* This code contributed by PrinciRaj1992 */

PHP

<?php 
// PHP program to implement Goldbach's  
// conjecture 
$MAX = 10000; 
  
// Array to store all prime less than 
// and equal to 10^6 
$primes = array(); 
  
// Utility function for Sieve of Sundaram 
function sieveSundaram() 
{ 
    global $MAX, $primes; 
      
    // In general Sieve of Sundaram, produces  
    // primes smaller than (2*x + 2) for a  
    // number given number x. Since we want 
    // primes smaller than MAX, we reduce  
    // MAX to half. This array is used to  
    // separate numbers of the form i + j + 2*i*j  
    // from others where 1 <= i <= j 
    $marked = array_fill(0, (int)($MAX / 2) +  
                                100, false); 
  
    // Main logic of Sundaram. Mark all  
    // numbers which do not generate prime 
    // number by doing 2*i+1 
    for ($i = 1; $i <= (sqrt($MAX) - 1) / 2; $i++) 
        for ($j = ($i * ($i + 1)) << 1;  
             $j <= $MAX / 2; $j = $j + 2 * $i + 1) 
            $marked[$j] = true; 
  
    // Since 2 is a prime number 
    array_push($primes, 2); 
  
    // Print other primes. Remaining primes  
    // are of the form 2*i + 1 such that  
    // marked[i] is false. 
    for ($i = 1; $i <= $MAX / 2; $i++) 
        if ($marked[$i] == false) 
            array_push($primes, 2 * $i + 1); 
} 
  
// Function to perform Goldbach's conjecture 
function findPrimes($n) 
{ 
    global $MAX, $primes; 
      
    // Return if number is not even  
    // or less than 3 
    if ($n <= 2 || $n % 2 != 0) 
    { 
        print("Invalid Input \n"); 
        return; 
    } 
  
    // Check only upto half of number 
    for ($i = 0; $primes[$i] <= $n / 2; $i++) 
    { 
        // find difference by subtracting  
        // current prime from n 
        $diff = $n - $primes[$i]; 
  
        // Search if the difference is also a  
        // prime number 
        if (in_array($diff, $primes)) 
        { 
            // Express as a sum of primes 
            print($primes[$i] . " + " . 
                  $diff . " = " . $n . "\n"); 
            return; 
        } 
    } 
} 
  
// Driver code 
  
// Finding all prime numbers before limit 
sieveSundaram(); 
  
// Express number as a sum of two primes 
findPrimes(4); 
findPrimes(38); 
findPrimes(100); 
  
// This code is contributed by chandan_jnu 
?> 

Output:

2 + 2 = 4
7 + 31 = 38
3 + 97 = 100

A Goldbach number is a positive integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach’s conjecture is that all even integers greater than 4 are Goldbach numbers.

References: Wiki
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My Personal Notes

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

C++

Java

Python3

C#

PHP

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