# 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.

1. Find the prime numbers using Sieve of Sundaram
2. Check if entered number is an even number greater than 2 or not, if no return.
3. 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 ` `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 primes = ``new` `ArrayList(); ` ` `  `// 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

 ` `

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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