Prime Triplet

Prime Triplet is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6). This is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself) except (2, 3, 5) and (3, 5, 7) .

Examples :

Input : n = 15
Output : 5 7 11
         7 11 13

Input : n = 25
Output : 5 7 11
         7 11 13
         11 13 17
         13 17 19
         17 19 23



A simple solution is to traverse through all numbers from 1 to n-6. For every number i check if i, i+2, i+6 or i, i+4, i+6 are primes. If yes, print triplet.

An efficient solution is to use Sieve of Eratosthenes to first find all prime numbers so that we can quickly check if a number is prime or not.

Below is the implementation of the approach.

C++

// C++ program to find prime triplets smaller
// than or equal to n.
#include <bits/stdc++.h>
using namespace std;
  
// function to detect prime number
// here we have used sieve method
// to detect prime number
void sieve(int n, bool prime[])
{
    for (int p = 2; p * p <= n; p++) {
  
        // If prime[p] is not changed, then it is a prime
        if (prime[p] == true) {
  
            // Update all multiples of p
            for (int i = p * 2; i <= n; i += p)
                prime[i] = false;
        }
    }
}
  
// function to print prime triplets
void printPrimeTriplets(int n)
{
    // Finding all primes from 1 to n
    bool prime[n + 1];
    memset(prime, true, sizeof(prime));
    sieve(n, prime);
      
    cout << "The prime triplets from 1 to " 
          << n << "are :" << endl;
    for (int i = 2; i <= n-6; ++i) {
  
        // triplets of form (p, p+2, p+6)
        if (prime[i] && prime[i + 2] && prime[i + 6])
            cout << i << " " << (i + 2) << " " << (i + 6) << endl;
  
        // triplets of form (p, p+4, p+6)
        else if (prime[i] && prime[i + 4] && prime[i + 6])
            cout << i << " " << (i + 4) << " " << (i + 6) << endl;
    }
}
  
int main()
{
    int n = 25;
    printPrimeTriplets(n);
    return 0;
}

Java

// Java program to find prime triplets
// smaller than or equal to n.
import java.io.*;
import java.util.*;
  
class GFG {
      
// function to detect prime number
// here we have used sieve method
// to detect prime number
    static void sieve(int n, boolean prime[])
    {
        for (int p = 2; p * p <= n; p++) {
      
            // If prime[p] is not changed,
            //then it is a prime
            if (prime[p] == true) {
      
                // Update all multiples of p
                for (int i = p * 2; i <= n; i += p)
                    prime[i] = false;
            }
        }
    }
      
    // function to print prime triplets
    static void printPrimeTriplets(int n)
    {
        // Finding all primes from 1 to n
        boolean prime[]=new boolean[n + 1];
        Arrays.fill(prime,true);
        sieve(n, prime);
          
        System.out.println("The prime triplets"+
                           " from 1 to " + n + "are :");
          
        for (int i = 2; i <= n-6; ++i) {
      
            // triplets of form (p, p+2, p+6)
            if (prime[i] && prime[i + 2] && prime[i + 6])
                System.out.println( i + " " + (i + 2) + 
                                    " " + (i + 6));
      
            // triplets of form (p, p+4, p+6)
            else if (prime[i] && prime[i + 4] && 
                     prime[i + 6])
                  
                System.out.println(i + " " + (i + 4) +
                                   " " + (i + 6));
        }
    }
      
    public static void main(String args[])
    {
        int n = 25;
        printPrimeTriplets(n);
    }
}
  
  
 /*This code is contributed by Nikita Tiwari.*/

Python3

# Python 3 program to find 
# prime triplets smaller
# than or equal to n.
  
# function to detect prime number
# using sieve method
# to detect prime number
def sieve(n, prime) :
      
    p = 2
      
    while (p * p <= n ) :
          
        # If prime[p] is not changed
        # , then it is a prime
        if (prime[p] == True) :
              
            # Update all multiples of p
            i = p * 2
          
            while ( i <= n ) :
                prime[i] = False
                i = i + p
          
        p = p + 1
          
  
# function to print 
# prime triplets
def printPrimeTriplets(n) :
  
    # Finding all primes 
    # from 1 to n
    prime = [True] * (n + 1)
    sieve(n, prime)
      
    print( "The prime triplets from 1 to ",
                               n , "are :")
      
    for i in range(2, n - 6 + 1) :
          
        # triplets of form (p, p+2, p+6)
        if (prime[i] and prime[i + 2] and
                            prime[i + 6]) :
            print( i , (i + 2) , (i + 6))
              
        # triplets of form (p, p+4, p+6)
        elif (prime[i] and prime[i + 4] and
                            prime[i + 6]) :
            print(i , (i + 4) , (i + 6))
              
# Driver code
n = 25
printPrimeTriplets(n)
  
# This code is contributed by Nikita Tiwari.

C#

// C# program to find prime 
// triplets smaller than or
// equal to n.
using System;
  
class GFG 
{
      
// function to detect 
// prime number
static void sieve(int n, 
                  bool[] prime)
{
    for (int p = 2; 
             p * p <= n; p++) 
    {
  
        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p] == false
        {
  
            // Update all multiples of p
            for (int i = p * 2; 
                     i <= n; i += p)
                prime[i] = true;
        }
    }
}
  
// function to print
// prime triplets
static void printPrimeTriplets(int n)
{
    // Finding all primes
    // from 1 to n
    bool[] prime = new bool[n + 1];
    sieve(n, prime);
      
    Console.WriteLine("The prime triplets "
                               "from 1 to "
                               n + " are :");
      
    for (int i = 2; i <= n - 6; ++i) 
    {
  
        // triplets of form (p, p+2, p+6)
        if (!prime[i] && 
            !prime[i + 2] && 
            !prime[i + 6])
            Console.WriteLine(i + " " + (i + 2) + 
                                  " " + (i + 6));
  
        // triplets of form (p, p+4, p+6)
        else if (!prime[i] && 
                 !prime[i + 4] && 
                 !prime[i + 6])
            Console.WriteLine(i + " " + (i + 4) + 
                                  " " + (i + 6));
    }
}
  
// Driver Code
public static void Main()
{
    int n = 25;
    printPrimeTriplets(n);
}
}
  
// This code is contributed by mits

PHP

<?php
// PHP program to find prime 
// triplets smaller than or
// equal to n.
  
// function to print 
// prime triplets
function printPrimeTriplets($n)
{
    // Finding all primes
    // from 1 to n
    $prime = array_fill(0, $n + 1, true);
      
    // to detect prime number
    for ($p = 2; $p * $p <= $n; $p++) 
    {
  
        // If prime[p] is not changed,
        // then it is a prime
        if ($prime[$p] == true) 
        {
  
            // Update all multiples of p
            for ($i = $p * 2; $i <= $n; $i += $p)
                $prime[$i] = false;
        }
    }
      
    echo "The prime triplets from 1 to "
                          $n . " are :\n";
    for ($i = 2; $i <= $n-6; ++$i
    {
  
        // triplets of form (p, p+2, p+6)
        if ($prime[$i] && $prime[$i + 2] && 
                          $prime[$i + 6])
            echo $i. " " . ($i + 2) . 
                     " " . ($i + 6) . "\n";
  
        // triplets of form (p, p+4, p+6)
        else if ($prime[$i] && $prime[$i + 4] && 
                               $prime[$i + 6])
            echo $i. " " . ($i + 4) . 
                     " " . ($i + 6) . "\n";
    
}
  
// Driver Code
$n = 25;
printPrimeTriplets($n);
  
// This code is contributed by mits.
?>

Output :

The prime triplets from 1 to 25 are :
5 7 11
7 11 13
11 13 17
13 17 19
17 19 23

References :
Wiki



My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.


Improved By : Mithun Kumar




Recommended Posts:



3 Average Difficulty : 3/5.0
Based on 1 vote(s)






User Actions