# Prime numbers after prime P with sum S

• Difficulty Level : Medium
• Last Updated : 06 May, 2021

Given three numbers sum S, prime P, and N, find all N prime numbers after prime P such that their sum is equal to S.
Examples :

```Input :  N = 2, P = 7, S = 28
Output : 11 17
Explanation : 11 and 17 are primes after
prime 7 and (11 + 17 = 28)

Input :  N = 3, P = 2, S = 23
Output : 3 7 13
5 7 11
Explanation : 3, 5, 7, 11 and 13 are primes
after prime 2. And (3 + 7 + 13 = 5 + 7 + 11
= 23)

Input :  N = 4, P = 3, S = 54
Output : 5 7 11 31
5 7 13 29
5 7 19 23
5 13 17 19
7 11 13 23
7 11 17 19
Explanation : All are prime numbers and
their sum is 54```

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Approach: The approach used is to produce all the primes less than S and greater than P. And then backtracking to find if such N primes exist whose sum equals S.
For example, S = 10, N = 2, P = 2 ## C++

 `// CPP Program to print all N primes after``// prime P whose sum equals S``#include ``#include ``#include ``using` `namespace` `std;` `// vector to store prime and N primes``// whose sum equals given S``vector<``int``> set;``vector<``int``> prime;` `// function to check prime number``bool` `isPrime(``int` `x)``{``    ``// square root of x``    ``int` `sqroot = ``sqrt``(x);``    ``bool` `flag = ``true``;` `    ``// since 1 is not prime number``    ``if` `(x == 1)``        ``return` `false``;` `    ``// if any factor is found return false``    ``for` `(``int` `i = 2; i <= sqroot; i++)``        ``if` `(x % i == 0)``            ``return` `false``;` `    ``// no factor found``    ``return` `true``;``}` `// function to display N primes whose sum equals S``void` `display()``{``    ``int` `length = set.size();``    ``for` `(``int` `i = 0; i < length; i++)``        ``cout << set[i] << ``" "``;``    ``cout << ``"\n"``;``}` `// function to evaluate all possible N primes``// whose sum equals S``void` `primeSum(``int` `total, ``int` `N, ``int` `S, ``int` `index)``{``    ``// if total equals S And``    ``// total is reached using N primes``    ``if` `(total == S && set.size() == N)``    ``{``        ``// display the N primes``        ``display();``        ``return``;``    ``}` `    ``// if total is greater than S``    ``// or if index has reached last element``    ``if` `(total > S || index == prime.size())``        ``return``;` `    ``// add prime[index] to set vector``    ``set.push_back(prime[index]);` `    ``// include the (index)th prime to total``    ``primeSum(total+prime[index], N, S, index+1);` `    ``// remove element from set vector``    ``set.pop_back();` `    ``// exclude (index)th prime``    ``primeSum(total, N, S, index+1);``}` `// function to generate all primes``void` `allPrime(``int` `N, ``int` `S, ``int` `P)``{``    ``// all primes less than S itself``    ``for` `(``int` `i = P+1; i <=S ; i++)``    ``{``        ``// if i is prime add it to prime vector``        ``if` `(isPrime(i))``            ``prime.push_back(i);``    ``}` `    ``// if primes are less than N``    ``if` `(prime.size() < N)``        ``return``;``    ``primeSum(0, N, S, 0);``}` `// Driver Code``int` `main()``{``    ``int` `S = 54, N = 2, P = 3;``    ``allPrime(N, S, P);``    ``return` `0;``}`

## Java

 `// Java Program to print``// all N primes after prime``// P whose sum equals S``import` `java.io.*;``import` `java.util.*;` `class` `GFG``{``    ``// vector to store prime``    ``// and N primes whose sum``    ``// equals given S``    ``static` `ArrayList set =``                     ``new` `ArrayList();``    ``static` `ArrayList prime =``                     ``new` `ArrayList();``    ` `    ``// function to check``    ``// prime number``    ``static` `boolean` `isPrime(``int` `x)``    ``{``        ``// square root of x``        ``int` `sqroot = (``int``)Math.sqrt(x);` `        ``// since 1 is not``        ``// prime number``        ``if` `(x == ``1``)``            ``return` `false``;``    ` `        ``// if any factor is``        ``// found return false``        ``for` `(``int` `i = ``2``;``                 ``i <= sqroot; i++)``            ``if` `(x % i == ``0``)``                ``return` `false``;``    ` `        ``// no factor found``        ``return` `true``;``    ``}``    ` `    ``// function to display N``    ``// primes whose sum equals S``    ``static` `void` `display()``    ``{``        ``int` `length = set.size();``        ``for` `(``int` `i = ``0``;``                 ``i < length; i++)``            ``System.out.print(``                   ``set.get(i) + ``" "``);``        ``System.out.println();``    ``}``    ` `    ``// function to evaluate``    ``// all possible N primes``    ``// whose sum equals S``    ``static` `void` `primeSum(``int` `total, ``int` `N,``                         ``int` `S, ``int` `index)``    ``{``        ``// if total equals S``        ``// And total is reached``        ``// using N primes``        ``if` `(total == S &&``            ``set.size() == N)``        ``{``            ``// display the N primes``            ``display();``            ``return``;``        ``}``    ` `        ``// if total is greater``        ``// than S or if index``        ``// has reached last``        ``// element``        ``// or if set size reached to maximum or greater than maximum``        ``if` `(total > S ||``            ``index == prime.size() || set.size() >= N)``            ``return``;``    ` `        ``// add prime.get(index)``        ``// to set vector``        ``set.add(prime.get(index));``    ` `        ``// include the (index)th``        ``// prime to total``        ``primeSum(total + prime.get(index),``                         ``N, S, index + ``1``);``    ` `        ``// remove element``        ``// from set vector``        ``set.remove(set.size() - ``1``);``    ` `        ``// exclude (index)th prime``        ``primeSum(total, N,``                 ``S, index + ``1``);``    ``}``    ` `    ``// function to generate``    ``// all primes``    ``static` `void` `allPrime(``int` `N,``                         ``int` `S, ``int` `P)``    ``{``        ``// all primes less``        ``// than S itself``        ``for` `(``int` `i = P + ``1``;``                 ``i <= S ; i++)``        ``{``            ``// if i is prime add``            ``// it to prime vector``            ``if` `(isPrime(i))``                ``prime.add(i);``        ``}``    ` `        ``// if primes are``        ``// less than N``        ``if` `(prime.size() < N)``            ``return``;``        ``primeSum(``0``, N, S, ``0``);``    ``}``    ` `    ``// Driver Code``    ``public` `static` `void` `main(String args[])``    ``{``        ``int` `S = ``54``, N = ``2``, P = ``3``;``        ``allPrime(N, S, P);``    ``}``}` `// This code is contributed by``// Manish Shaw(manishshaw1)`

## Python3

 `# Python Program to print``# all N primes after prime``# P whose sum equals S``import` `math` `# vector to store prime``# and N primes whose``# sum equals given S``set` `=` `[]``prime ``=` `[]` `# function to``# check prime number``def` `isPrime(x) :` `    ``# square root of x``    ``sqroot ``=` `int``(math.sqrt(x))``    ``flag ``=` `True` `    ``# since 1 is not``    ``# prime number``    ``if` `(x ``=``=` `1``) :``        ``return` `False` `    ``# if any factor is``    ``# found return false``    ``for` `i ``in` `range``(``2``, sqroot ``+` `1``) :``        ``if` `(x ``%` `i ``=``=` `0``) :``            ``return` `False` `    ``# no factor found``    ``return` `True` `# function to display N``# primes whose sum equals S``def` `display() :` `    ``global` `set``, prime``    ``length ``=` `len``(``set``)``    ``for` `i ``in` `range``(``0``, length) :``        ``print` `(``set``[i], end ``=` `" "``)``    ``print` `()` `# function to evaluate``# all possible N primes``# whose sum equals S``def` `primeSum(total, N,``             ``S, index) :``    ` `    ``global` `set``, prime``    ` `    ``# if total equals S``    ``# And total is reached``    ``# using N primes``    ``if` `(total ``=``=` `S ``and``         ``len``(``set``) ``=``=` `N) :``    ` `        ``# display the N primes``        ``display()``        ``return` `    ``# if total is greater``    ``# than S or if index``    ``# has reached last element``    ``if` `(total > S ``or``        ``index ``=``=` `len``(prime)) :``        ``return` `    ``# add prime[index]``    ``# to set vector``    ``set``.append(prime[index])` `    ``# include the (index)th``    ``# prime to total``    ``primeSum(total ``+` `prime[index],``                  ``N, S, index ``+` `1``)` `    ``# remove element``    ``# from set vector``    ``set``.pop()` `    ``# exclude (index)th prime``    ``primeSum(total, N,``             ``S, index ``+` `1``)` `# function to generate``# all primes``def` `allPrime(N, S, P) :` `    ``global` `set``, prime``    ` `    ``# all primes less``    ``# than S itself``    ``for` `i ``in` `range``(P ``+` `1``,``                   ``S ``+` `1``) :``    ` `        ``# if i is prime add``        ``# it to prime vector``        ``if` `(isPrime(i)) :``            ``prime.append(i)``    ` `    ``# if primes are``    ``# less than N``    ``if` `(``len``(prime) < N) :``        ``return``    ``primeSum(``0``, N, S, ``0``)` `# Driver Code``S ``=` `54``N ``=` `2``P ``=` `3``allPrime(N, S, P)` `# This code is contributed by``# Manish Shaw(manishshaw1)`

## C#

 `// C# Program to print all``// N primes after prime P``// whose sum equals S``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{``    ``// vector to store prime``    ``// and N primes whose sum``    ``// equals given S``    ``static` `List<``int``> ``set` `= ``new` `List<``int``>();``    ``static` `List<``int``> prime = ``new` `List<``int``>();``    ` `    ``// function to check prime number``    ``static` `bool` `isPrime(``int` `x)``    ``{``        ``// square root of x``        ``int` `sqroot = (``int``)Math.Sqrt(x);` `        ``// since 1 is not prime number``        ``if` `(x == 1)``            ``return` `false``;``    ` `        ``// if any factor is``        ``// found return false``        ``for` `(``int` `i = 2; i <= sqroot; i++)``            ``if` `(x % i == 0)``                ``return` `false``;``    ` `        ``// no factor found``        ``return` `true``;``    ``}``    ` `    ``// function to display N``    ``// primes whose sum equals S``    ``static` `void` `display()``    ``{``        ``int` `length = ``set``.Count;``        ``for` `(``int` `i = 0; i < length; i++)``            ``Console.Write(``set``[i] + ``" "``);``        ``Console.WriteLine();``    ``}``    ` `    ``// function to evaluate``    ``// all possible N primes``    ``// whose sum equals S``    ``static` `void` `primeSum(``int` `total, ``int` `N,``                         ``int` `S, ``int` `index)``    ``{``        ``// if total equals S And``        ``// total is reached using N primes``        ``if` `(total == S && ``set``.Count == N)``        ``{``            ``// display the N primes``            ``display();``            ``return``;``        ``}``    ` `        ``// if total is greater than``        ``// S or if index has reached``        ``// last element``        ``if` `(total > S || index == prime.Count)``            ``return``;``    ` `        ``// add prime[index]``        ``// to set vector``        ``set``.Add(prime[index]);``    ` `        ``// include the (index)th``        ``// prime to total``        ``primeSum(total + prime[index],``                         ``N, S, index + 1);``    ` `        ``// remove element``        ``// from set vector``        ``set``.RemoveAt(``set``.Count - 1);``    ` `        ``// exclude (index)th prime``        ``primeSum(total, N, S, index + 1);``    ``}``    ` `    ``// function to generate``    ``// all primes``    ``static` `void` `allPrime(``int` `N,``                         ``int` `S, ``int` `P)``    ``{``        ``// all primes less than S itself``        ``for` `(``int` `i = P + 1; i <=S ; i++)``        ``{``            ``// if i is prime add``            ``// it to prime vector``            ``if` `(isPrime(i))``                ``prime.Add(i);``        ``}``    ` `        ``// if primes are``        ``// less than N``        ``if` `(prime.Count < N)``            ``return``;``        ``primeSum(0, N, S, 0);``    ``}``    ` `    ``// Driver Code``    ``static` `void` `Main()``    ``{``        ``int` `S = 54, N = 2, P = 3;``        ``allPrime(N, S, P);``    ``}``}` `// This code is contributed by``// Manish Shaw(manishshaw1)`

## PHP

 ` ``\$S` `||``        ``\$index` `== ``count``(``\$prime``))``        ``return``;` `    ``// add prime[index]``    ``// to set vector``    ``array_push``(``\$set``,``               ``\$prime``[``\$index``]);` `    ``// include the (index)th``    ``// prime to total``    ``primeSum(``\$total` `+ ``\$prime``[``\$index``],``             ``\$N``, ``\$S``, ``\$index` `+ 1);` `    ``// remove element``    ``// from set vector``    ``array_pop``(``\$set``);` `    ``// exclude (index)th prime``    ``primeSum(``\$total``, ``\$N``, ``\$S``,``             ``\$index` `+ 1);``}` `// function to generate``// all primes``function` `allPrime(``\$N``, ``\$S``, ``\$P``)``{``    ``global` `\$set``, ``\$prime``;``    ` `    ``// all primes less``    ``// than S itself``    ``for` `(``\$i` `= ``\$P` `+ 1;``         ``\$i` `<= ``\$S` `; ``\$i``++)``    ``{``        ``// if i is prime add``        ``// it to prime vector``        ``if` `(isPrime(``\$i``))``            ``array_push``(``\$prime``, ``\$i``);``    ``}` `    ``// if primes are``    ``// less than N``    ``if` `(``count``(``\$prime``) < ``\$N``)``        ``return``;``    ``primeSum(0, ``\$N``, ``\$S``, 0);``}` `// Driver Code``\$S` `= 54; ``\$N` `= 2; ``\$P` `= 3;``allPrime(``\$N``, ``\$S``, ``\$P``);` `// This code is contributed by``// Manish Shaw(manishshaw1)``?>`

## Javascript

 ``
Output:
```7 47
11 43
13 41
17 37
23 31```

Optimizations :
The above solution can be optimized by pre-computing all required primes using Sieve of Eratosthenes

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