Cunningham chain

A Cunningham chain is a sequence of prime numbers. It is of 2 types: 

• Cunningham chain of the first kind: It is a sequence of prime numbers of length n described as below :
  

Let p1, p2, p3, …., p_n be a cunningham chain of length n than 
p₂ = 2*p₁ + 1 
p₃ = 4*p₁ + 3 
p₄ = 8*p₁ + 7 
. . . 
. . . 
p_n = 2^n-1*p1 + (2^n-1 – 1)

• Here p1, p2, p3, …., pn are all prime numbers. If any value of p comes out to be non-prime then chain ends at the number which came before it.
  for p0 = 2, the sequence will be 2 5 11 23 47
  Below is the implementation of the above: 
  

C++

// C++ program for cunningham chain
// Function to print the series
// of first kind
#include <bits/stdc++.h>

using namespace std;

// Function to print
// Cunningham chain of the first kind
void print(int p0)
{
    int p1, i = 0, x, flag, k;

    // Iterate till all elements
    // are printed
    while (1) {
        flag = 1;
        x = (int)(pow(2, i));
        p1 = x * p0 + (x - 1);

        // check prime or not
        for (k = 2; k < p1; k++) {
            if (p1 % k == 0) {
                flag = 0;
                break;
            }
        }
        if (flag == 0)
            break;
        printf("%d ", p1);
        i++;
    }
}

// Driver Code
int main()
{
    int p0 = 2;
    print(p0);

    return 0;
}

Java

// Java Program to print the
// series of first kind
class GFG
{

// Function to print
// Cunningham chain
// of the first kind
static void print(int p0)
{
    int p1, i = 0, x, flag, k;

    // Iterate till all
    // elements are printed
    while (true)
    {
        flag = 1;
        x = (int)(Math.pow(2, i));
        p1 = x * p0 + (x - 1);

        // check prime or not
        for (k = 2; k < p1; k++)
        {
            if (p1 % k == 0)
            {
                flag = 0;
                break;
            }
        }
        if (flag == 0)
            break;
        System.out.print(" " + p1);
        i++;
    }
}

// Driver Code
public static void main(String args[])
{
    int p0 = 2;
    print(p0);
}
}

// This code is contributed
// by Kirti_Mangal

Python3

# Python3 program for cunningham chain

# Function to print Cunningham chain
# of the first kind
def print_C(p0):

    i = 0;

    # Iterate till all elements
    # are printed
    while(True):
        flag = 1;
        x = pow(2, i);
        p1 = x * p0 + (x - 1);

        # check prime or not
        for k in range(2, p1):
            if (p1 % k == 0):
                flag = 0;
                break;

        if (flag == 0):
            break;

        print(p1, end = " ");
        i += 1;

# Driver Code
p0 = 2;
print_C(p0);

# This code is contributed by mits

C#

// C# Program to print the
// series of first kind
using System;
class GFG
{

// Function to print
// Cunningham chain
// of the first kind
static void print(int p0)
{
    int p1, i = 0, x, flag, k;

    // Iterate till all
    // elements are printed
    while (true)
    {
        flag = 1;
        x = (int)(Math.Pow(2, i));
        p1 = x * p0 + (x - 1);

        // check prime or not
        for (k = 2; k < p1; k++)
        {
            if (p1 % k == 0)
            {
                flag = 0;
                break;
            }
        }
        if (flag == 0)
            break;
        Console.Write(" " + p1);
        i++;
    }
}

// Driver Code
public static void Main()
{
    int p0 = 2;
    print(p0);
}
}

// This code is contributed
// by Akanksha Rai(Abby_akku)

PHP

<?php
// PHP program for cunningham chain
// Function to print
// Cunningham chain of the first kind
function print_C($p0)
{
    $p1 = 0; $i = 0; $x; $flag; $k;

    // Iterate till all elements
    // are printed
    while (1)
    {
        $flag = 1;
        $x = pow(2, $i);
        $p1 = $x * $p0 + ($x - 1);

        // check prime or not
        for ($k = 2; $k < $p1; $k++)
        {
            if ($p1 % $k == 0) {
                $flag = 0;
                break;
            }
        }
        if ($flag == 0)
            break;
        echo $p1 . " ";
        $i++;
    }
}

// Driver Code
$p0 = 2;
print_C($p0);

// This code is contributed
// by Akanksha Rai(Abby_akku)

Javascript

<script>
// Javascript program for cunningham chain
// Function to print
// Cunningham chain of the first kind
function print_C(p0)
{
    let p1 = 0;
    let i = 0;
    let x;
    let flag;
    let k;

    // Iterate till all elements
    // are printed
    while (1)
    {
        flag = 1;
        x = Math.pow(2, i);
        p1 = x * p0 + (x - 1);

        // check prime or not
        for (let k = 2; k < p1; k++)
        {
            if (p1 % k == 0) {
                flag = 0;
                break;
            }
        }
        if (flag == 0)
            break;
        document.write(p1 + " ");
        i++;
    }
}

// Driver Code
let p0 = 2;
print_C(p0);

// This code is contributed
// by gfgking

</script>

Output:

2 5 11 23 47

• Cunningham chain of the second kind: It is a sequence of prime numbers of length n described as below:
  

Let p1, p2, p3, …., p_n be a cunningham chain of length n than 
p₂ = 2*p₁ – 1 
p₃ = 4*p₁ – 3 
p₄ = 8*p₁ – 7 
. . . 
. . . 
p_n = 2^n-1*p1 – (2^n-1 – 1)

• for p0 = 19, the sequence will be 19, 37, 73.
  Below is the implementation of the above: 
  

C++

// C++ program for cunningham chain
// Function to print the series
// of second kind
#include <bits/stdc++.h>

using namespace std;

// Function to print
// Cunningham chain of the second kind
void print(int p0)
{
    int p1, i = 0, x, flag, k;

    // Iterate till all elements
    // are printed
    while (1) {
        flag = 1;
        x = (int)(pow(2, i));
        p1 = x * p0 - (x - 1);

        // check prime or not
        for (k = 2; k < p1; k++) {
            if (p1 % k == 0) {
                flag = 0;
                break;
            }
        }
        if (flag == 0)
            break;
        printf("%d ", p1);
        i++;
    }
}

// Driver Code
int main()
{
    int p0 = 19;
    print(p0);

    return 0;
}

Java

// Java program for cunningham chain
// Function to print the series
// of second kind

class GFG{

// Function to print Cunningham chain
//  of the second kind
static void print(int p0)
{
    int p1, i = 0, x, flag, k;

    // Iterate till all elements
    // are printed
    while (true)
    {
        flag = 1;
        x = (int)(Math.pow(2, i));
        p1 = x * p0 - (x - 1);

        // check prime or not
        for (k = 2; k < p1; k++)
        {
            if (p1 % k == 0)
            {
                flag = 0;
                break;
            }
        }
        if (flag == 0)
            break;
        System.out.print(p1+" ");
        i++;
    }
}

// Driver Code
public static void main(String[] args)
{
    int p0 = 19;
    print(p0);
}
}

// This code is contributed by mits

Python3

# Python3 program for cunningham chain

# Function to print Cunningham chain
# of the second kind
def print_t(p0):

    i = 0;

    # Iterate till all elements
    # are printed
    while (True):
        flag = 1;
        x = pow(2, i);
        p1 = x * p0 - (x - 1);

        # check prime or not
        for k in range(2, p1):
            if (p1 % k == 0):
                flag = 0;
                break;

        if (flag == 0):
            break;
        print(p1,end=" ");
        i+=1;

# Driver Code
p0 = 19;
print_t(p0);

# This code is contributed by mits

C#

// C# program for cunningham chain
// Function to print the series
// of second kind
using System;
class GFG
{

// Function to print
// Cunningham chain of the second kind
static void print(int p0)
{
    int p1, i = 0, x, flag, k;

    // Iterate till all elements
    // are printed
    while (true)
    {
        flag = 1;
        x = (int)(Math.Pow(2, i));
        p1 = x * p0 - (x - 1);

        // check prime or not
        for (k = 2; k < p1; k++)
        {
            if (p1 % k == 0)
            {
                flag = 0;
                break;
            }
        }
        if (flag == 0)
            break;
        Console.Write(p1 + " ");
        i++;
    }
}

// Driver Code
static void Main()
{
    int p0 = 19;
    print(p0);
}
}

// This code is contributed by mits

PHP

<?php
// PHP program for cunningham chain


// Function to print
// Cunningham chain of the second kind
function print_t($p0)
{
    $p1; $i = 0; $x; $flag; $k;

    // Iterate till all elements
    // are printed
    while (1)
    {
        $flag = 1;
        $x = pow(2, $i);
        $p1 = $x * $p0 - ($x - 1);

        // check prime or not
        for ($k = 2; $k < $p1; $k++) {
            if ($p1 % $k == 0) {
                $flag = 0;
                break;
            }
        }
        if ($flag == 0)
            break;
        echo $p1 . " ";
        $i++;
    }
}

// Driver Code
$p0 = 19;
print_t($p0);

// This code is contributed
// by Akanksha Rai(Abby_akku)

Javascript

<script>

// JavaScript program for cunningham chain

// Function to print
// Cunningham chain of the second kind
function print(p0)
{
    var p1, i = 0, x, flag = 1, k, m = 4;

    // Iterate till all elements
    // are printed
    while (flag) {
        flag = 1;
        x = Math.pow(2, i);
        p1 = x * p0 - (x - 1);

        // check prime or not
        for (k = 2; k < p1; k++) {
            if (p1 % k == 0) {
                flag = 0;
                break;
            }
        }
        if (flag == 0)
            break;
        document.write(p1 + " ");
        i++;
    }
}

// Driver Code
var p0 = 19;
print(p0);

//This code is contributed by Shivani

</script>

Output:

19 37 73

Time complexity : O(n^2), where n is the number of elements in the Cunningham chain of the first kind.

Space complexity: O(1), as it only uses a few variables and doesn’t increase with the size of the input.