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# 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```

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```

The time complexity of this algorithm is O(n^2) where n is the number of prime numbers between P and S. This is because the nested for loop in the allPrime() function runs for O(n) and the recursive primeSum() function runs for O(n) time as well.

The space complexity is O(n) because we are using a vector of size n to store prime numbers and another vector of size n to store a possible combination of N primes.

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

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