Find Largest Special Prime which is less than or equal to a given number

Given a number N. The task is to find the largest special prime which is less than or equal to N.

A special prime is a number which can be created by placing digits one after another such the all the resulting numbers are prime.

Examples:

Input : N = 379
Output : 379
Explanation: 379 can be created as => 3 => 37 => 379
Here, all the numbers ie. 3, 37, 379 are prime.

Input : N = 100
Output : 79
Explanation: 79 can be created as => 7 => 79, 
where both 7, 79 are prime numbers.


Approach: The idea is to use Sieve Of eratosthenes. Build the sieve array up to the number N. Then start iteratively back from the number N checking if the number is prime. If it is prime then check if it is special prime or not.

Now, to check if a number is a special prime or not. Keep dividing the number by 10 and at each point check whether the remaining number is prime or not, which we can do by referring our Sieve array which we have built.

Below is the implementation of the above approach:

C++

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// CPP program to find the Largest Special Prime
// which is less than or equal to a given number
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to check whether the number
// is a special prime or not
bool checkSpecialPrime(bool* sieve, int num)
{
    // While number is not equal to zero
    while (num) {
        // If the number is not prime
        // return false.
        if (!sieve[num]) {
            return false;
        }
  
        // Else remove the last digit
        // by dividing the number by 10.
        num /= 10;
    }
  
    // If the number has become zero
    // then the number is special prime,
    // hence return true
    return true;
}
  
// Function to find the Largest Special Prime
// which is less than or equal to a given number
void findSpecialPrime(int N)
{
    bool sieve[N + 10];
  
    // Initially all numbers are considered Primes.
    memset(sieve, true, sizeof(sieve));
    sieve[0] = sieve[1] = false
    for (long long i = 2; i <= N; i++) {
        if (sieve[i]) {
  
            for (long long j = i * i; j <= N; j += i) {
                sieve[j] = false;
            }
        }
    }
  
    // There is always an answer possible
    while (true) {
        // Checking if the number is a
        // special prime or not
        if (checkSpecialPrime(sieve, N)) {
            // If yes print the number
            // and break the loop.
            cout << N << '\n';
            break;
        }
        // Else decrement the number.
        else
            N--;
    }
}
  
// Driver code
int main()
{
    findSpecialPrime(379);
    findSpecialPrime(100);
  
    return 0;
}

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Java

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// Java program to find the Largest Special Prime
// which is less than or equal to a given number
  
class GFG
{
  
        // Function to check whether the number
        // is a special prime or not
    static boolean checkSpecialPrime(boolean [] sieve, int num)
        {
            // While number is not equal to zero
            while (num!=0) {
                // If the number is not prime
                // return false.
                if (!sieve[num]) {
                    return false;
                }
          
                // Else remove the last digit
                // by dividing the number by 10.
                num /= 10;
            }
          
            // If the number has become zero
            // then the number is special prime,
            // hence return true
            return true;
        }
          
        // Function to find the Largest Special Prime
        // which is less than or equal to a given number
        static void findSpecialPrime(int N)
        {
            boolean []sieve=new boolean[N+10];
            sieve[0] = sieve[1] = false;
  
            // Initially all numbers are considered Primes.
            for(int i=0;i<N+10;i++)
                sieve[i]=true;
              
            for (int i = 2; i <= N; i++) {
                if (sieve[i]) {
          
                    for ( int j = i * i; j <= N; j += i) {
                        sieve[j] = false;
                    }
                }
            }
          
            // There is always an answer possible
            while (true) {
                // Checking if the number is a
                // special prime or not
                if (checkSpecialPrime(sieve, N)) {
                    // If yes print the number
                    // and break the loop.
                    System.out.println(N);
                    break;
                }
                // Else decrement the number.
                else
                    N--;
            }
        }
          
        // Driver code
        public static void main(String [] args)
        {
            findSpecialPrime(379);
            findSpecialPrime(100);
          
              
        }
  
// This code is contributed by ihritik
  
}

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Python 3

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# Python 3 program to find the Largest 
# Special Prime which is less than or
# equal to a given number
  
# Function to check whether the number
# is a special prime or not
def checkSpecialPrime(sieve, num):
  
    # While number is not equal to zero
    while (num) :
          
        # If the number is not prime
        # return false.
        if (not sieve[num]) :
            return False
  
        # Else remove the last digit
        # by dividing the number by 10.
        num //= 10
  
    # If the number has become zero
    # then the number is special prime,
    # hence return true
    return True
  
# Function to find the Largest Special 
# Prime which is less than or equal to 
# a given number
def findSpecialPrime(N):
  
    # Initially all numbers are
    # considered Primes.
    sieve = [True] * (N + 10)
    sieve[0] = sieve[1] = False;
    for i in range(2, N + 1) :
        if (sieve[i]) :
  
            for j in range(i * i, N + 1, i) :
                sieve[j] = False
  
    # There is always an answer possible
    while (True) :
          
        # Checking if the number is 
        # a special prime or not
        if (checkSpecialPrime(sieve, N)):
              
            # If yes print the number
            # and break the loop.
            print( N) 
            break
              
        # Else decrement the number.
        else:
            N -= 1
  
# Driver code
if __name__ == "__main__":
    findSpecialPrime(379)
    findSpecialPrime(100)
  
# This code is contributed 
# by ChitraNayal

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C#

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// C# program to find the Largest Special Prime
// which is less than or equal to a given number
  
using System;
class GFG
{
  
        // Function to check whether the number
        // is a special prime or not
    static bool checkSpecialPrime(bool [] sieve, int num)
        {
            // While number is not equal to zero
            while (num!=0) {
                // If the number is not prime
                // return false.
                if (!sieve[num]) {
                    return false;
                }
          
                // Else remove the last digit
                // by dividing the number by 10.
                num /= 10;
            }
          
            // If the number has become zero
            // then the number is special prime,
            // hence return true
            return true;
        }
          
        // Function to find the Largest Special Prime
        // which is less than or equal to a given number
        static void findSpecialPrime(int N)
        {
            bool []sieve=new bool[N+10];
              
              
            // Initially all numbers are considered Primes.
            for(int i = 0; i < N + 10; i++)
                sieve[i] = true;
                  
            sieve[0] = sieve[1] = false;
            for (int i = 2; i <= N; i++) {
                if (sieve[i]) {
          
                    for ( int j = i * i; j <= N; j += i) {
                        sieve[j] = false;
                    }
                }
            }
          
            // There is always an answer possible
            while (true) {
                // Checking if the number is a
                // special prime or not
                if (checkSpecialPrime(sieve, N)) {
                    // If yes print the number
                    // and break the loop.
                    Console.WriteLine(N);
                    break;
                }
                // Else decrement the number.
                else
                    N--;
            }
        }
          
        // Driver code
        public static void Main()
        {
            findSpecialPrime(379);
            findSpecialPrime(100);
          
              
        }
  
// This code is contributed by ihritik
  
}

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PHP

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<?php
// PHP program to find the Largest 
// Special Prime which is less than
// or equal to a given number
  
// Function to check whether the number
// is a special prime or not
function checkSpecialPrime(&$sieve, $num)
{
    // While number is not equal to zero
    while ($num
    {
        // If the number is not prime
        // return false.
        if (!$sieve[$num])
        {
            return false;
        }
  
        // Else remove the last digit
        // by dividing the number by 10.
        $num = (int)($num / 10);
    }
  
    // If the number has become zero
    // then the number is special prime,
    // hence return true
    return true;
}
  
// Function to find the Largest Special Prime
// which is less than or equal to a given number
function findSpecialPrime($N)
{
  
    // Initially all numbers are 
    // considered Primes.
    $sieve = array_fill(0, $N + 10, true);
    $sieve[0] = $sieve[1] = false;
      
    for ($i = 2; $i <= $N; $i++) 
    {
        if ($sieve[$i]) 
        {
            for ($j = $i * $i; $j <= $N; $j += $i
            {
                $sieve[$j] = false;
            }
        }
    }
  
    // There is always an answer possible
    while (true) 
    {
          
        // Checking if the number is a
        // special prime or not
        if (checkSpecialPrime($sieve, $N))
        {
            // If yes print the number
            // and break the loop.
              
            echo $N . "\n";
            break;
        }
          
        // Else decrement the number.
        else
            $N--;
    }
}
  
// Driver code
findSpecialPrime(379);
findSpecialPrime(100);
  
// This code is contributed by mits
?>

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Output:

379
79

Time Complexity: O(N*log(log N))



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