# Find three prime numbers with given sum

Given an integer N, the task is to find three prime numbers X, Y and Z such that the sum of these three numbers is equal to N i.e. X + Y + Z = N.

Examples:

Input: N = 20
Output: 2 5 13

Input: N = 34
Output: 2 3 29

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

• Generate prime numbers using Sieve of Eratosthenes
• Start from the first prime number.
• Take another number from the generated list.
• Subtract first number and second number from the original number to obtain the third number.
• Check if the third number is a prime number.
• If the third number is a prime number then output the three numbers.
• Otherwise, repeat the process for the second number and consequently the first number
• If the answer does not exist then print -1.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` ` `  `#include ` `using` `namespace` `std; ` ` `  `const` `int` `MAX = 100001; ` ` `  `// The vector primes holds ` `// the prime numbers ` `vector<``int``> primes; ` ` `  `// Function to generate prime numbers ` `void` `initialize() ` `{ ` ` `  `    ``// Initialize the array elements to 0s ` `    ``bool` `numbers[MAX] = {}; ` `    ``int` `n = MAX; ` `    ``for` `(``int` `i = 2; i * i <= n; i++) ` `        ``if` `(!numbers[i]) ` `            ``for` `(``int` `j = i * i; j <= n; j += i) ` ` `  `                ``// Set the non-primes to true ` `                ``numbers[j] = ``true``; ` ` `  `    ``// Fill the vector primes with prime ` `    ``// numbers which are marked as false ` `    ``// in the numbers array ` `    ``for` `(``int` `i = 2; i <= n; i++) ` `        ``if` `(numbers[i] == ``false``) ` `            ``primes.push_back(i); ` `} ` ` `  `// Function to print three prime numbers ` `// which sum up to the number N ` `void` `findNums(``int` `num) ` `{ ` ` `  `    ``bool` `ans = ``false``; ` `    ``int` `first = -1, second = -1, third = -1; ` `    ``for` `(``int` `i = 0; i < num; i++) { ` ` `  `        ``// Take the first prime number ` `        ``first = primes[i]; ` `        ``for` `(``int` `j = 0; j < num; j++) { ` ` `  `            ``// Take the second prime number ` `            ``second = primes[j]; ` ` `  `            ``// Subtract the two prime numbers ` `            ``// from the N to obtain the third number ` `            ``third = num - first - second; ` ` `  `            ``// If the third number is prime ` `            ``if` `(binary_search(primes.begin(), ` `                              ``primes.end(), third)) { ` `                ``ans = ``true``; ` `                ``break``; ` `            ``} ` `        ``} ` `        ``if` `(ans) ` `            ``break``; ` `    ``} ` `    ``// Print the three prime numbers ` `    ``// if the solution exists ` `    ``if` `(ans) ` `        ``cout << first << ``" "` `             ``<< second << ``" "` `<< third << endl; ` `    ``else` `        ``cout << -1 << endl; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `n = 101; ` ` `  `    ``// Function for generating prime numbers ` `    ``// using Sieve of Eratosthenes ` `    ``initialize(); ` ` `  `    ``// Function to print the three prime ` `    ``// numbers whose sum is equal to N ` `    ``findNums(n); ` ` `  `    ``return` `0; ` `} `

## Python3

 `# Python3 implementation of the approach  ` `from` `math ``import` `sqrt ` ` `  `MAX` `=` `100001``;  ` ` `  `# The vector primes holds  ` `# the prime numbers  ` `primes ``=` `[];  ` ` `  `# Function to generate prime numbers  ` `def` `initialize() : ` ` `  `    ``# Initialize the array elements to 0s  ` `    ``numbers ``=` `[``0``]``*``(``MAX` `+` `1``); ` `    ``n ``=` `MAX``;  ` `    ``for` `i ``in` `range``(``2``, ``int``(sqrt(n)) ``+` `1``) :  ` `        ``if` `(``not` `numbers[i]) : ` `            ``for` `j ``in` `range``( i ``*` `i , n ``+` `1``, i) :  ` ` `  `                ``# Set the non-primes to true  ` `                ``numbers[j] ``=` `True``;  ` ` `  `    ``# Fill the vector primes with prime  ` `    ``# numbers which are marked as false  ` `    ``# in the numbers array  ` `    ``for` `i ``in` `range``(``2``, n ``+` `1``) : ` `        ``if` `(numbers[i] ``=``=` `False``) : ` `            ``primes.append(i);  ` ` `  `# Function to print three prime numbers  ` `# which sum up to the number N  ` `def` `findNums(num) :  ` ` `  `    ``ans ``=` `False``;  ` `    ``first ``=` `-``1``;  ` `    ``second ``=` `-``1``; ` `    ``third ``=` `-``1``;  ` `    ``for` `i ``in` `range``(num) : ` ` `  `        ``# Take the first prime number  ` `        ``first ``=` `primes[i];  ` `        ``for` `j ``in` `range``(num) : ` ` `  `            ``# Take the second prime number  ` `            ``second ``=` `primes[j];  ` ` `  `            ``# Subtract the two prime numbers  ` `            ``# from the N to obtain the third number  ` `            ``third ``=` `num ``-` `first ``-` `second;  ` ` `  `            ``# If the third number is prime  ` `            ``if` `(third ``in` `primes) : ` `                ``ans ``=` `True``;  ` `                ``break``;  ` `     `  `        ``if` `(ans) : ` `            ``break``;  ` `     `  `    ``# Print the three prime numbers  ` `    ``# if the solution exists  ` `    ``if` `(ans) : ` `        ``print``(first , second , third);  ` `    ``else` `: ` `        ``print``(``-``1``);  ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"` `:  ` ` `  `    ``n ``=` `101``;  ` ` `  `    ``# Function for generating prime numbers  ` `    ``# using Sieve of Eratosthenes  ` `    ``initialize();  ` ` `  `    ``# Function to print the three prime  ` `    ``# numbers whose sum is equal to N  ` `    ``findNums(n);  ` ` `  `# This code is contributed by AnkitRai01 `

Output:

```2 2 97
```

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Improved By : AnkitRai01