Find the smallest twins in given range

2.6

Given a range [low..high], print the smallest twin numbers in given range (low and high inclusive). Two numbers are twins if they are primes and there difference is 2.

Example:

Input:  low = 10,  high = 100
Output: Smallest twins in given range: (11, 13)
Both 11 and 13 are prime numbers and difference 
between them is two, therefore twins.  And these
are the smallest twins in [10..100]

Input:  low = 50,  high = 100
Output: Smallest twins in given range: (59, 61) 

A Simple Solution is to start to start from low and for every number x check if x and x + 2 are primes are not. Here x varies from low to high-2.

An Efficient Solution is to use Sieve of Eratosthenes

1) Create a boolean array "prime[0..high]" and initialize all 
   entries in it as true. A value in prime[i] will finally 
   be false if i is not a prime number, else true.

2) Run a loop from p = 2 to high. 
    a) If prime[p] is true, then p is prime. [See this]
    b) Mark all multiples of p as not prime in prime[]. 

3) Run a loop from low to high and print the first twins
   using prime[] built in step 2.   

Below is C++ implementation of above idea.

// C++ program to find the smallest twin in given range
#include <bits/stdc++.h>
using namespace std;

void printTwins(int low, int high)
{
    // Create a boolean array "prime[0..high]" and initialize
    // all entries it as true. A value in prime[i] will finally
    // be false if i is Not a prime, else true.
    bool prime[high+1];
    memset(prime, true, sizeof(prime));

    prime[0] = prime[1] = false;

    // Look for the smallest twin
    for (int p=2; p<=floor(sqrt(high))+1; p++)
    {
        // If p is not marked, then it is a prime
        if (prime[p])
        {
            // Update all multiples of p
            for (int i=p*2; i<=high; i += p)
                prime[i] = false;
        }
    }

    // Now print the smallest twin in range
    for (int i=low; i<=high; i++)
    {
        if (prime[i] && prime[i+2])
        {
            cout << "Smallest twins in given range: ("
                 << i << ", " << i+2 <<  ")";
            break;
        }
    }
}

// Driver program
int main()
{
    printTwins(10, 100);
    return 0;
}

Output:

Smallest twins in given range: (11, 13)

Thanks to Utkarsh Trivedi for suggesting this solution.

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