# Prime numbers after prime P with sum S

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
```

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

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) ` `?> `

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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