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Find three prime numbers with given sum
• Last Updated : 05 Jan, 2021

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

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;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;``import` `java.lang.*;``import` `java.io.*;` `class` `GFG{` `static` `int` `MAX = ``100001``;``  ` `// The vector primes holds``// the prime numbers``static` `ArrayList primes;``  ` `// Function to generate prime numbers``static` `void` `initialize()``{``    ` `    ``// Initialize the array elements to 0s``    ``boolean``[] numbers = ``new` `boolean``[MAX + ``1``];``    ``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.add(i);``}``  ` `// Function to print three prime numbers``// which sum up to the number N``static` `void` `findNums(``int` `num)``{``    ``boolean` `ans = ``false``;``    ``int` `first = -``1``, second = -``1``, third = -``1``;``    ` `    ``for``(``int` `i = ``0``; i < num; i++)``    ``{``        ` `        ``// Take the first prime number``        ``first = primes.get(i);``        ` `        ``for``(``int` `j = ``0``; j < num; j++)``        ``{``            ` `            ``// Take the second prime number``            ``second = primes.get(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` `(Collections.binarySearch(``                ``primes, third) >= ``0``)``            ``{``                ``ans = ``true``;``                ``break``;``            ``}``        ``}``        ``if` `(ans)``            ``break``;``    ``}``    ` `    ``// Print the three prime numbers``    ``// if the solution exists``    ``if` `(ans)``       ``System.out.println(first + ``" "` `+``                          ``second + ``" "` `+``                          ``third);``    ``else``        ``System.out.println(-``1` `);``}` `// Driver code``public` `static` `void` `main (String[] args)``{``    ``int` `n = ``101``;``    ``primes = ``new` `ArrayList<>();``    ` `    ``// 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 offbeat`

## 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`

## C#

 `// C# implementation of the approach``using` `System;``using` `System.Collections.Generic;``class` `GFG``{  ``  ``static` `int` `MAX = 100001;` `  ``// The vector primes holds``  ``// the prime numbers``  ``static` `List<``int``> primes = ``new` `List<``int``>();` `  ``// Function to generate prime numbers``  ``static` `void` `initialize()``  ``{` `    ``// Initialize the array elements to 0s``    ``bool``[] numbers = ``new` `bool``[MAX + 1];``    ``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.Add(i);``  ``}` `  ``// Function to print three prime numbers``  ``// which sum up to the number N``  ``static` `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` `(Array.BinarySearch(primes.ToArray(), third) >= 0)``        ``{``          ``ans = ``true``;``          ``break``;``        ``}``      ``}``      ``if` `(ans)``        ``break``;``    ``}` `    ``// Print the three prime numbers``    ``// if the solution exists``    ``if` `(ans)``      ``Console.WriteLine(first + ``" "` `+ second + ``" "` `+ third);``    ``else``      ``Console.WriteLine(-1);``  ``}` `  ``// Driver code``  ``static` `void` `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);``  ``}``}` `// This code is contributed by divyesh072019`
Output:
`2 2 97`

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