Sixy Primes

Given two integers l and r representing a range [l, r], find all Sixy prime pairs within the range. Two prime numbers are called Sixy primes if their difference is exactly 6.

Return all such pairs in increasing order of the first prime. For every valid pair (p, p + 6), append p followed by p + 6 to the result. If no such pair exists, return an empty list.

Examples:

Input: l = 11, r = 19
Output: [11, 17, 13, 19]
Explanation: There are total two pair possible with difference 6 and these are 11,17,13,19.
Input: l = 6, r = 20
Output: [7, 13, 11, 17, 13, 19]
Explanation: There are total three pair possible with difference 6 and these are 7,13,11,17,13,19.

Try It Yourself

Table of Content

[Naive Approach] By Checking Every Pair - O((r - l + 1) * sqrt(r)) Time and O(1) Space
[Expected Approach] Using Sieve of Eratosthenes - O(r log log r) Time and O(r) Space

[Naive Approach] By Checking Every Pair - O((r - l + 1) * sqrt(r)) Time and O(1) Space

The idea is to simply iterate through every number from l to r - 6 and check whether both i and i + 6 are prime. If both are prime, then (i, i + 6) is a valid Sixy prime pair.

Create an empty result vector.
Traverse every number i from l to r - 6.
Check if i and i + 6 are prime.
If both are prime, append i and i + 6 to the result.

C++

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

// Returns true if n is a prime number.
bool isPrime(int n)
{
    // Numbers less than 2 are not prime.
    if (n < 2)
        return false;

    // Check for factors from 2 to sqrt(n).
    for (int i = 2; i * i <= n; i++)
    {
        if (n % i == 0)
            return false;
    }

    return true;
}

// Returns all Sixy prime pairs in the range [l, r].
vector<int> sixyPrime(int l, int r)
{
    vector<int> res;

    // Check every possible pair (i, i + 6).
    for (int i = l; i <= r - 6; i++)
    {
        // If both numbers are prime, store the pair.
        if (isPrime(i) && isPrime(i + 6))
        {
            res.push_back(i);
            res.push_back(i + 6);
        }
    }

    return res;
}

int main()
{
    int l = 11, r = 19;

    vector<int> ans = sixyPrime(l, r);
    for (int i = 0; i < ans.size(); i += 1)
        cout << ans[i] << " ";

    return 0;
}

Java

import java.util.*;

class GFG {
    
    // Returns true if n is a prime number.
    static boolean isPrime(int n)
    {
        // Numbers less than 2 are not prime.
        if (n < 2)
            return false;

        // Check for factors from 2 to sqrt(n).
        for (int i = 2; i * i <= n; i++) {
            if (n % i == 0)
                return false;
        }

        return true;
    }

    // Returns all Sixy prime pairs in the range [l, r].
    static ArrayList<Integer> sixyPrime(int l, int r)
    {
        ArrayList<Integer> res = new ArrayList<>();

        // Check every possible pair (i, i + 6).
        for (int i = l; i <= r - 6; i++) {

            // If both numbers are prime, store the pair.
            if (isPrime(i) && isPrime(i + 6)) {
                res.add(i);
                res.add(i + 6);
            }
        }

        return res;
    }

    public static void main(String[] args)
    {
        int l = 11, r = 19;

        ArrayList<Integer> ans = sixyPrime(l, r);

        for (int x : ans)
            System.out.print(x + " ");
    }
}

Python

def isPrime(n):

    # Numbers less than 2 are not prime.
    if n < 2:
        return False

    # Check for factors from 2 to sqrt(n).
    i = 2
    while i * i <= n:
        if n % i == 0:
            return False
        i += 1

    return True


# Returns all Sixy prime pairs in the range [l, r].
def sixyPrime(l, r):
    res = []

    # Check every possible pair (i, i + 6).
    for i in range(l, r - 5):

        # If both numbers are prime, store the pair.
        if isPrime(i) and isPrime(i + 6):
            res.append(i)
            res.append(i + 6)

    return res


# Driver Code
if __name__ == "__main__":
    l = 11
    r = 19

    ans = sixyPrime(l, r)

    for x in ans:
        print(x, end=" ")

using System;
using System.Collections.Generic;

class GFG {
    // Returns true if n is a prime number.
    static bool IsPrime(int n)
    {
        // Numbers less than 2 are not prime.
        if (n < 2)
            return false;

        // Check for factors from 2 to sqrt(n).
        for (int i = 2; i * i <= n; i++) {
            if (n % i == 0)
                return false;
        }

        return true;
    }

    // Returns all Sixy prime pairs in the range [l, r].
    static List<int> sixyPrime(int l, int r)
    {
        List<int> res = new List<int>();

        // Check every possible pair (i, i + 6).
        for (int i = l; i <= r - 6; i++) {
            
            // If both numbers are prime, store the pair.
            if (IsPrime(i) && IsPrime(i + 6)) {
                res.Add(i);
                res.Add(i + 6);
            }
        }

        return res;
    }

    static void Main()
    {
        int l = 11, r = 19;

        List<int> ans = sixyPrime(l, r);

        foreach(int x in ans) Console.Write(x + " ");
    }
}

JavaScript

// Returns true if n is a prime number.
function isPrime(n)
{

    // Numbers less than 2 are not prime.
    if (n < 2)
        return false;

    // Check for factors from 2 to sqrt(n).
    for (let i = 2; i * i <= n; i++) {
        if (n % i === 0)
            return false;
    }

    return true;
}

// Returns all Sixy prime pairs in the range [l, r].
function sixyPrime(l, r)
{
    let res = [];

    // Check every possible pair (i, i + 6).
    for (let i = l; i <= r - 6; i++) {

        // If both numbers are prime, store the pair.
        if (isPrime(i) && isPrime(i + 6)) {
            res.push(i);
            res.push(i + 6);
        }
    }

    return res;
}
// Driver Code
let l = 11;
let r = 19;

let ans = sixyPrime(l, r);

for (let x of ans)
    process.stdout.write(x + " ");

Output

11 17 13 19

[Expected Approach] Using Sieve of Eratosthenes - O(r log log r) Time and O(r) Space

We can check primality of every number from 2 to r in one preprocessing step using the Sieve of Eratosthenes. Once the sieve is built, checking whether a number is prime takes O(1) time. We then simply traverse the range [l, r - 6] and collect every pair (i, i + 6) where both numbers are prime.

Create a boolean array prime of size r + 1 and initialize all entries as true.
Mark 0 and 1 as non-prime.
Traverse from 2 to sqrt(r).
If the current number is prime, mark all of its multiples starting from i * i as non-prime.
Create an empty result vector. Traverse from max(l, 2) to r - 6.
If both prime[i] and prime[i + 6] are true, append i and i + 6 to the result. Return the result.

C++

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

// Returns all Sixy prime pairs in the range [l, r].
vector<int> sixyPrime(int l, int r)
{
    vector<int> res;

    // There are no prime numbers less than 2.
    if (r < 2)
        return res;

    // Stores whether each number is prime.
    vector<bool> prime(r + 1, true);

    // Mark 0 and 1 as non-prime.
    prime[0] = false;
    prime[1] = false;

    // Generate all prime numbers up to r using Sieve of Eratosthenes.
    for (int i = 2; i * i <= r; i++)
    {
        if (prime[i])
        {
            // Mark all multiples of i as non-prime.
            for (int j = i * i; j <= r; j += i)
                prime[j] = false;
        }
    }

    // Check every possible pair (i, i + 6).
    for (int i = max(l, 2); i <= r - 6; i++)
    {
        // If both numbers are prime, store the pair.
        if (prime[i] && prime[i + 6])
        {
            res.push_back(i);
            res.push_back(i + 6);
        }
    }

    return res;
}

int main()
{
    int l = 11, r = 19;

    vector<int> ans = sixyPrime(l, r);

    for (int x : ans)
        cout << x << " ";

    return 0;
}