Prime triplets consisting of values up to N having difference between two elements equal to the third

Given a positive integer N, the task is to find all the prime triplets {P, Q, R} such that P = R – Q and P, Q and R is less than N.

Examples:

Input: N = 8
Output:
2 3 5
2 5 7
Explanation:
The only 2 prime triplets satisfying the given conditions are:

• {2, 3, 5}: P = 2, Q = 3, R = 5. Therefore, P, Q and R are prime numbers and P = R – Q.
• {2, 5, 7}: P = 2, Q = 5, R = 7. Therefore, P, Q and R are prime numbers and P = R – Q.

Input: N = 5
Output: 2 3 5

Approach: The given problem can be solved based on the following observations:

• By rearranging the given equation, it can be observed that P + Q = R, and the sum of two odd numbers is even and the sum of one odd and one even is odd.
• As there is only one even prime i.e., 2. Let P is odd prime and Q is odd prime then R can never be a prime number, so it is necessary that P should be always 2 which is even prime and Q is odd prime then R should be an odd prime. So there is necessary to find prime Q such that Q > 2 and R = P + Q ≤ N (where P = 2) and R should be prime.

Follow the steps below to solve the problem:

• Precompute all the prime numbers over the range [1, N] using Sieve of Eratosthenes.
• Initialize a vector of vectors, say V, to stores all the resultant triplets.
• Iterate over the range [3, N] using a variable, say i. If i and (i + 2) are prime and 2 + i ≤ N, then store the current triplets in the vector V.
• After completing the above steps, print all the triplets stored in V[].

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Stores 1 and 0 at indices which
// are prime and non-prime respectively
bool prime[100000];

// Function to find all prime
// numbers from the range [0, N]
void SieveOfEratosthenes(int n)
{
    // Consider all numbers to prime initially
    memset(prime, true, sizeof(prime));

    // Iterate over the range [2, sqrt(N)]
    for (int p = 2; p * p <= n; p++) {

        // If p is a prime
        if (prime[p] == true) {

            // Update all tultiples
            // of p as false
            for (int i = p * p;
                 i <= n; i += p) {
                prime[i] = false;
            }
        }
    }
}

// Function to find all prime triplets
// satisfying the given conditions
void findTriplets(int N)
{
    // Generate all primes up to N
    SieveOfEratosthenes(N);

    // Stores the triplets
    vector<vector<int> > V;

    // Iterate over the range [3, N]
    for (int i = 3; i <= N; i++) {

        // Check for the condition
        if (2 + i <= N && prime[i]
            && prime[2 + i]) {

            // Store the triplets
            V.push_back({ 2, i, i + 2 });
        }
    }

    // Print all the stored triplets
    for (int i = 0; i < V.size(); i++) {
        cout << V[i][0] << " "
             << V[i][1] << " "
             << V[i][2] << "\n";
    }
}

// Driver Code
int main()
{
    int N = 8;
    findTriplets(N);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;

class GFG{

// Stores 1 and 0 at indices which
// are prime and non-prime respectively
static boolean[] prime = new boolean[100000];

static void initialize()
{
    for(int i = 0; i < 100000; i++)
        prime[i] = true;
}

// Function to find all prime
// numbers from the range [0, N]
static void SieveOfEratosthenes(int n)
{

    // Iterate over the range [2, sqrt(N)]
    for(int p = 2; p * p <= n; p++)
    {

        // If p is a prime
        if (prime[p] == true)
        {

            // Update all tultiples
            // of p as false
            for(int i = p * p; i <= n; i += p)
            {
                prime[i] = false;
            }
        }
    }
}

// Function to find all prime triplets
// satisfying the given conditions
static void findTriplets(int N)
{

    // Generate all primes up to N
    SieveOfEratosthenes(N);

    // Stores the triplets
    ArrayList<ArrayList<Integer>> V = new ArrayList<ArrayList<Integer>>();
    // List<List<int> > V = new List<List<int>>();

    // Iterate over the range [3, N]
    for(int i = 3; i <= N; i++)
    {

        // Check for the condition
        if (2 + i <= N && prime[i] && prime[2 + i])
        {

            // Store the triplets
            ArrayList<Integer> a1 = new ArrayList<Integer>();
            a1.add(2);
            a1.add(i);
            a1.add(i + 2);
            V.add(a1);
        }
    }

    // Print all the stored triplets
    for(int i = 0; i < V.size(); i++)
    {
        System.out.println(V.get(i).get(0) + " " +
                           V.get(i).get(1) + " " +
                           V.get(i).get(2));
    }
}

// Driver Code
public static void main(String args[])
{
    initialize();
    int N = 8;

    findTriplets(N);
}
}

// This code is contributed by ipg2016107

Python3

# Python3 program for the above approach
from math import sqrt

# Stores 1 and 0 at indices which
# are prime and non-prime respectively
prime = [True for i in range(100000)]

# Function to find all prime
# numbers from the range [0, N]
def SieveOfEratosthenes(n):

    # Iterate over the range [2, sqrt(N)]
    for p in range(2, int(sqrt(n)) + 1, 1):

        # If p is a prime
        if (prime[p] == True):

            # Update all tultiples
            # of p as false
            for i in range(p * p, n + 1, p):
                prime[i] = False

# Function to find all prime triplets
# satisfying the given conditions
def findTriplets(N):

    # Generate all primes up to N
    SieveOfEratosthenes(N)

    # Stores the triplets
    V = []

    # Iterate over the range [3, N]
    for i in range(3, N + 1, 1):

        # Check for the condition
        if (2 + i <= N and prime[i] and prime[2 + i]):

            # Store the triplets
            V.append([2, i, i + 2])

    # Print all the stored triplets
    for i in range(len(V)):
        print(V[i][0], V[i][1], V[i][2])

# Driver Code
if __name__ == '__main__':
    N = 8
    findTriplets(N)

    # This code is contributed by bgangwar59.

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

// Stores 1 and 0 at indices which
// are prime and non-prime respectively
static bool[] prime = new bool[100000];

static void initialize()
{
    for(int i = 0; i < 100000; i++)
        prime[i] = true;
}

// Function to find all prime
// numbers from the range [0, N]
static void SieveOfEratosthenes(int n)
{

    // Iterate over the range [2, sqrt(N)]
    for(int p = 2; p * p <= n; p++)
    {

        // If p is a prime
        if (prime[p] == true)
        {

            // Update all tultiples
            // of p as false
            for(int i = p * p; i <= n; i += p)
            {
                prime[i] = false;
            }
        }
    }
}

// Function to find all prime triplets
// satisfying the given conditions
static void findTriplets(int N)
{

    // Generate all primes up to N
    SieveOfEratosthenes(N);

    // Stores the triplets
    List<List<int>> V = new List<List<int>>();

    // Iterate over the range [3, N]
    for(int i = 3; i <= N; i++)
    {

        // Check for the condition
        if (2 + i <= N && prime[i] ==
                  true && prime[2 + i])
        {

            // Store the triplets
            List<int> a1 = new List<int>();
            a1.Add(2);
            a1.Add(i);
            a1.Add(i + 2);
            V.Add(a1);
        }
    }

    // Print all the stored triplets
    for(int i = 0; i < V.Count; i++)
    {
        Console.WriteLine(V[i][0] + " " +
                          V[i][1] + " " +
                          V[i][2]);
    }
}

// Driver Code
public static void Main()
{
    initialize();
    int N = 8;

    findTriplets(N);
}
}

// This code is contributed by SURENDRA_GANGWAR

Javascript

<script>

// Javascript program for the above approach

// Stores 1 and 0 at indices which
// are prime and non-prime respectively
var prime = new Array(100000);

// Function to find all prime
// numbers from the range [0, N]
function SieveOfEratosthenes(n)
{

    // Consider all numbers to prime initially
    prime.fill(true);

    // Iterate over the range [2, sqrt(N)]
    for(var p = 2; p * p <= n; p++)
    {

        // If p is a prime
        if (prime[p] == true)
        {

            // Update all tultiples
            // of p as false
            for(var i = p * p;
                    i <= n; i += p)
            {
                prime[i] = false;
            }
        }
    }
}

// Function to find all prime triplets
// satisfying the given conditions
function findTriplets(N)
{

    // Generate all primes up to N
    SieveOfEratosthenes(N);

    // Stores the triplets
    var V = [];

    // Iterate over the range [3, N]
    for(var i = 3; i <= N; i++)
    {

        // Check for the condition
        if (2 + i <= N && prime[i] == true &&
                          prime[2 + i] == true)
        {

            // Store the triplets
            var a1 = [2, i, i + 2];

            V.push(a1);
        }
    }

    // Print all the stored triplets
    for(var i = 0; i < V.length; i++)
    {
        document.write(V[i][0] + " " +
                       V[i][1] + " " +
                       V[i][2] + "<br>");
    }
}

// Driver code
N = 8;

findTriplets(N);

// This code is contributed by SoumikMondal

</script>

Output:

2 3 5
2 5 7

Time Complexity: O(N*log(log(N)))
Auxiliary Space: O(1)