Modify array by merging elements with addition such that it consists of only Primes.

Given an array arr[] consisting of positive integers, the task is to check whether we can modify the array by adding any of the elements of the array such that it consists of only Primes.
Examples:

Input: arr[] = {3, 5, 7, 21}
Output: YES
Explanation:
So the array becomes {7, 29} which consists of only primes.

Input: {2, 5, 12, 16}
Output: NO
Explanation:
There is no possible combination among elements which will make the array containing all primes

Input: {3, 5, 13, 22}
Output: YES
Explanation:
Only one combination is possible,
Add all elements – 3+5+13+22 = {43} which is only prime

Approach:

• We can solve this problem using Dynamic Programming with Bitmasks. We can represent our DP state as a mask which is subset of elements.
• So let our dp array be DP[mask] which represents whether upto this mask (the chosen elements) the subset formed will elements are primes only.
• The max number of bits in mask will be the number of elements in the array.
• We keep on marking the elements encoded as a sequence in mask (If i-th index element is selected, then i-th bit is set in the mask) and keep checking whether the current chosen element (current sum) is prime, if it is prime and all other elements are visited, then we return true, and we get the answer.
• Else if other elements are not visited, then as the current sum is prime, we just need to search for other elements which can individually or by summing up to form some primes, so we can safely mark our current sum as 0 again.
• If the mask is full (All elements are visited) and the current sum is not prime, we return false, because there is atleast one sum that is not prime.
• Recurrence:
```DP[mask] = solve(curr + arr[i], mask | 1<<i)
where 0 <= i < n
```

Below is the implementation of the above approach.

C++

 `// C++ program to find the  ` `// array of primes ` `#include ` `using` `namespace` `std; ` ` `  `// DP array to store the ` `// ans for max 20 elements ` `bool` `dp[1 << 20]; ` ` `  `// To check whether the  ` `// number is prime or not ` `bool` `isprime(``int` `n) ` `{ ` `    ``if` `(n == 1) ` `        ``return` `false``; ` `    ``for` `(``int` `i = 2; i * i <= n; i++) ` `    ``{ ` `        ``if` `(n % i == 0) { ` `            ``return` `false``; ` `        ``} ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function to check whether the ` `// array can be modify so that  ` `// there are only primes ` `int` `solve(``int` `arr[], ``int` `curr,  ` `          ``int` `mask, ``int` `n) ` `{ ` ` `  `    ``// If curr is prime and all ` `    ``// elements are visited, ` `    ``// return true ` `    ``if` `(isprime(curr)) ` `    ``{ ` `        ``if` `(mask == (1 << n) - 1) ` `        ``{ ` `            ``return` `true``; ` `        ``} ` ` `  `        ``// If all elements are not  ` `        ``// visited, set curr=0, to  ` `        ``// search for new prime sum ` `        ``curr = 0; ` `    ``} ` ` `  `    ``// If all elements are visited ` `    ``if` `(mask == (1 << n) - 1) ` `    ``{ ` ` `  `        ``// If the current sum is ` `        ``// not prime return false ` `        ``if` `(!isprime(curr)) ` `        ``{ ` `            ``return` `false``; ` `        ``} ` `    ``} ` ` `  `    ``// If this state is already  ` `    ``// calculated, return the ` `    ``// answer directly ` `    ``if` `(dp[mask]) ` `        ``return` `dp[mask]; ` ` `  `    ``// Try all state of mask ` `    ``for` `(``int` `i = 0; i < n; i++) ` `    ``{ ` `        ``// If ith index is not set ` `        ``if` `(!(mask & 1 << i)) ` `        ``{ ` `            ``// Add the current element ` `            ``// and set ith index and recur ` `            ``if` `(solve(arr, curr + arr[i] ` `                      ``, mask | 1 << i, n)) ` `            ``{ ` `                ``// If subset can be formed ` `                ``// then return true ` `                ``return` `true``; ` `            ``} ` `        ``} ` `    ``} ` `     `  `    ``// After every possibility of mask,  ` `    ``// if the subset is not formed, ` `    ``// return false by memoizing. ` `    ``return` `dp[mask] = ``false``; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``int` `arr[] = { 3, 6, 7, 13 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]); ` `     `  `    ``if``(solve(arr, 0, 0, n)) ` `    ``{ ` `        ``cout << ``"YES"``; ` `    ``} ` `    ``else` `    ``{ ` `         ``cout << ``"NO"``; ` `    ``} ` `    ``return` `0; ` `    `  `} `

Java

 `// Java program to find the array of primes ` `import` `java.util.*; ` ` `  `class` `GFG{ ` ` `  `// dp array to store the ` `// ans for max 20 elements ` `static` `boolean` `[]dp = ``new` `boolean``[``1` `<< ``20``]; ` ` `  `// To check whether the  ` `// number is prime or not ` `static` `boolean` `isprime(``int` `n) ` `{ ` `    ``if` `(n == ``1``) ` `        ``return` `false``; ` ` `  `    ``for``(``int` `i = ``2``; i * i <= n; i++) ` `    ``{ ` `       ``if` `(n % i == ``0``) ` `       ``{ ` `           ``return` `false``; ` `       ``} ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function to check whether the ` `// array can be modify so that  ` `// there are only primes ` `static` `boolean` `solve(``int` `arr[], ``int` `curr,  ` `                     ``int` `mask, ``int` `n) ` `{ ` ` `  `    ``// If curr is prime and all ` `    ``// elements are visited, ` `    ``// return true ` `    ``if` `(isprime(curr)) ` `    ``{ ` `        ``if` `(mask == (``1` `<< n) - ``1``) ` `        ``{ ` `            ``return` `true``; ` `        ``} ` ` `  `        ``// If all elements are not  ` `        ``// visited, set curr=0, to  ` `        ``// search for new prime sum ` `        ``curr = ``0``; ` `    ``} ` ` `  `    ``// If all elements are visited ` `    ``if` `(mask == (``1` `<< n) - ``1``) ` `    ``{ ` ` `  `        ``// If the current sum is ` `        ``// not prime return false ` `        ``if` `(!isprime(curr)) ` `        ``{ ` `            ``return` `false``; ` `        ``} ` `    ``} ` ` `  `    ``// If this state is already  ` `    ``// calculated, return the ` `    ``// answer directly ` `    ``if` `(dp[mask]) ` `        ``return` `dp[mask]; ` ` `  `    ``// Try all state of mask ` `    ``for``(``int` `i = ``0``; i < n; i++) ` `    ``{ ` `        `  `       ``// If ith index is not set ` `       ``if` `((mask & (``1` `<< i)) == ``0``) ` `       ``{ ` `            `  `           ``// Add the current element ` `           ``// and set ith index and recur ` `           ``if` `(solve(arr, curr + arr[i],  ` `                     ``mask | ``1` `<< i, n)) ` `           ``{ ` `                `  `               ``// If subset can be formed ` `               ``// then return true ` `               ``return` `true``; ` `           ``} ` `       ``} ` `    ``} ` `     `  `    ``// After every possibility of mask,  ` `    ``// if the subset is not formed, ` `    ``// return false by memoizing. ` `    ``return` `dp[mask] = ``false``; ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `arr[] = { ``3``, ``6``, ``7``, ``13` `}; ` `    ``int` `n = arr.length; ` `     `  `    ``if``(solve(arr, ``0``, ``0``, n)) ` `    ``{ ` `        ``System.out.print(``"YES"``); ` `    ``} ` `    ``else` `    ``{ ` `        ``System.out.print(``"NO"``); ` `    ``} ` `} ` `} ` ` `  `// This code is contributed by Rohit_ranjan `

Python3

 `# Python3 program to find the array ` `# of primes  ` ` `  `# DP array to store the  ` `# ans for max 20 elements  ` `dp ``=` `[``0``] ``*` `(``1` `<< ``20``) ` ` `  `# To check whether the  ` `# number is prime or not  ` `def` `isprime(n):  ` `     `  `    ``if` `(n ``=``=` `1``):  ` `        ``return` `False` `    ``for` `i ``in` `range``(``2``, n ``+` `1``):  ` `        ``if` `(n ``%` `i ``=``=` `0``):  ` `            ``return` `False` `     `  `    ``return` `True` ` `  `# Function to check whether the  ` `# array can be modify so that  ` `# there are only primes  ` `def` `solve(arr, curr, mask, n):  ` ` `  `    ``# If curr is prime and all  ` `    ``# elements are visited,  ` `    ``# return true  ` `    ``if` `(isprime(curr)):  ` `        ``if` `(mask ``=``=` `(``1` `<< n) ``-` `1``):  ` `            ``return` `True` `         `  `        ``# If all elements are not  ` `        ``# visited, set curr=0, to  ` `        ``# search for new prime sum  ` `        ``curr ``=` `0` `     `  `    ``# If all elements are visited  ` `    ``if` `(mask ``=``=` `(``1` `<< n) ``-` `1``):  ` ` `  `        ``# If the current sum is  ` `        ``# not prime return false  ` `        ``if` `(isprime(curr) ``=``=` `False``):  ` `            ``return` `False` `         `  `    ``# If this state is already  ` `    ``# calculated, return the  ` `    ``# answer directly  ` `    ``if` `(dp[mask] !``=` `False``): ` `        ``return` `dp[mask]  ` ` `  `    ``# Try all state of mask  ` `    ``for` `i ``in` `range``(n): ` `         `  `        ``# If ith index is not set  ` `        ``if` `((mask & ``1` `<< i) ``=``=` `False``): ` `             `  `            ``# Add the current element  ` `            ``# and set ith index and recur  ` `            ``if` `(solve(arr, curr ``+` `arr[i], ` `                           ``mask | ``1` `<< i, n)):  ` `                                `  `                ``# If subset can be formed  ` `                ``# then return true  ` `                ``return` `True` `                 `  `    ``# After every possibility of mask,  ` `    ``# if the subset is not formed,  ` `    ``# return false by memoizing.  ` `    ``return` `(dp[mask] ``=``=` `False``) ` ` `  `# Driver code  ` `arr ``=` `[ ``3``, ``6``, ``7``, ``13` `]  ` ` `  `n ``=` `len``(arr)  ` `     `  `if` `(solve(arr, ``0``, ``0``, n)):  ` `    ``print``(``"YES"``) ` `else``:  ` `    ``print``(``"NO"``) ` `     `  `# This code is contributed by code_hunt `

C#

 `// C# program to find the array of primes ` `using` `System; ` ` `  `class` `GFG{ ` ` `  `// dp array to store the ` `// ans for max 20 elements ` `static` `bool` `[]dp = ``new` `bool``[1 << 20]; ` ` `  `// To check whether the  ` `// number is prime or not ` `static` `bool` `isprime(``int` `n) ` `{ ` `    ``if` `(n == 1) ` `        ``return` `false``; ` ` `  `    ``for``(``int` `i = 2; i * i <= n; i++) ` `    ``{ ` `       ``if` `(n % i == 0) ` `       ``{ ` `           ``return` `false``; ` `       ``} ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `// Function to check whether the ` `// array can be modify so that  ` `// there are only primes ` `static` `bool` `solve(``int` `[]arr, ``int` `curr,  ` `                  ``int` `mask, ``int` `n) ` `{ ` ` `  `    ``// If curr is prime and all ` `    ``// elements are visited, ` `    ``// return true ` `    ``if` `(isprime(curr)) ` `    ``{ ` `        ``if` `(mask == (1 << n) - 1) ` `        ``{ ` `            ``return` `true``; ` `        ``} ` ` `  `        ``// If all elements are not  ` `        ``// visited, set curr=0, to  ` `        ``// search for new prime sum ` `        ``curr = 0; ` `    ``} ` ` `  `    ``// If all elements are visited ` `    ``if` `(mask == (1 << n) - 1) ` `    ``{ ` ` `  `        ``// If the current sum is ` `        ``// not prime return false ` `        ``if` `(!isprime(curr)) ` `        ``{ ` `            ``return` `false``; ` `        ``} ` `    ``} ` ` `  `    ``// If this state is already  ` `    ``// calculated, return the ` `    ``// answer directly ` `    ``if` `(dp[mask]) ` `        ``return` `dp[mask]; ` ` `  `    ``// Try all state of mask ` `    ``for``(``int` `i = 0; i < n; i++) ` `    ``{ ` `        `  `       ``// If ith index is not set ` `       ``if` `((mask & (1 << i)) == 0) ` `       ``{ ` `            `  `           ``// Add the current element ` `           ``// and set ith index and recur ` `           ``if` `(solve(arr, curr + arr[i],  ` `                     ``mask | 1 << i, n)) ` `           ``{ ` `                `  `               ``// If subset can be formed ` `               ``// then return true ` `               ``return` `true``; ` `           ``} ` `       ``} ` `    ``} ` `     `  `    ``// After every possibility of mask,  ` `    ``// if the subset is not formed, ` `    ``// return false by memoizing. ` `    ``return` `dp[mask] = ``false``; ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `[]arr = { 3, 6, 7, 13 }; ` `    ``int` `n = arr.Length; ` `     `  `    ``if``(solve(arr, 0, 0, n)) ` `    ``{ ` `        ``Console.Write(``"YES"``); ` `    ``} ` `    ``else` `    ``{ ` `        ``Console.Write(``"NO"``); ` `    ``} ` `} ` `} ` ` `  `// This code is contributed by Rohit_ranjan `

Output:

```YES
```

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