Primality Test | Set 5(Using Lucas-Lehmer Series)

In this article we will discuss about Lucas-Lehmer series which is used to check primality of prime numbers of the form 2p – 1 where p is an integer.

First let’s see what is Lucas-Lehmer series.

The Lucas-Lehmer series can be expressed as :

     \[s_i = \left \{   \begin{tabular}{c}   4 \hspace{1.4cm} when i = 0;  \\   s_{i-1}^2 - 2 \hspace{.5cm}   otherwise.   \end{tabular}  \]

Hence the series is:
Term 0: 4,
Term 1: 4*4 – 2 = 14,
Term 2: 14*14 – 2 = 194,
Term 3: 194*194 – 2 = 37634,
Term 4: 37634*37634 – 2 = 1416317954, … and so on.



Below is the program to find out first n terms of the Lucas-Lehmer series.

C++

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// C++ program to find out Lucas-Lehmer series.
#include <iostream>
#include <vector>
using namespace std;
  
// Function to find out first n terms
// (considering 4 as 0th term) of
// Lucas-Lehmer series.
void LucasLehmer(int n) {
  
  // the 0th term of the series is 4.
  unsigned long long current_val = 4;
  
  // create an array to store the terms.
  vector<unsigned long long> series;
  
  // compute each term and add it to the array.
  series.push_back(current_val);
  for (int i = 0; i < n; i++) {
    current_val = current_val * current_val - 2;
    series.push_back(current_val);
  }
  
  // print out the terms one by one.
  for (int i = 0; i <= n; i++) 
    cout << "Term " << i << ": "
        << series[i] << endl;  
}
  
// Driver program
int main() {
  int n = 5;
  LucasLehmer(n);
  return 0;
}

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Java

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// Java program to find out
// Lucas-Lehmer series.
import java.util.*;
  
class GFG 
{
  
    // Function to find out 
    // first n terms(considering 
    // 4 as 0th term) of Lucas-
    // Lehmer series.
    static void LucasLehmer(int n) 
    {
  
        // the 0th term of
        // the series is 4.
        long current_val = 4;
  
        // create an array
        // to store the terms.
        ArrayList<Long> series = new ArrayList<>();
  
         // compute each term 
        // and add it to the array.
        series.add(current_val);
        for (int i = 0; i < n; i++) 
        {
            current_val = current_val
                    * current_val - 2;
            series.add(current_val);
        }
  
        // print out the
       // terms one by one.
        for (int i = 0; i <= n; i++) 
        {
            System.out.println("Term " + i
                    + ": " + series.get(i));
        }
    }
  
    // Driver Code
    public static void main(String[] args) 
    {
  
        int n = 5;
        LucasLehmer(n);
    }
}
  
// This code has been contributed by 29AjayKumar

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C#

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// C# program to find out
// Lucas-Lehmer series.
using System;
using System.Collections.Generic;
  
class GFG
{
      
// Function to find out 
// first n terms(considering 
// 4 as 0th term) of Lucas-
// Lehmer series.
static void LucasLehmer(int n) 
{
  
// the 0th term of
// the series is 4.
long current_val = 4;
  
// create an array
// to store the terms.
List<long> series = new List<long>();
  
// compute each term 
// and add it to the array.
series.Add(current_val);
for (int i = 0; i < n; i++)
{
    current_val = current_val * 
                  current_val - 2;
    series.Add(current_val);
}
  
// print out the
// terms one by one.
for (int i = 0; i <= n; i++) 
    Console.WriteLine("Term " + i + 
                      ": " + series[i]); 
}
  
// Driver Code
static void Main()
{
    int n = 5;
    LucasLehmer(n);
}
}
  
// This code is contributed by 
// ManishShaw(manishshaw1)

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Output:

Term 0: 4
Term 1: 14
Term 2: 194
Term 3: 37634
Term 4: 1416317954
Term 5: 2005956546822746114

We can use string to store the big numbers of the series.

Now what is the relation with prime numbers of this Lucas-Lehmer series?

1. First thing is that we can only check the primality of those numbers which we can represent as, x = (2p – 1) where p is an integer.
2. Now we have to find out the (p-1)th term of Lucas-Lehmer series.
3. If this term is a multiple of x, then x is a prime number.
4. When x is large, i.e. p is large then we may find difficulties to find out the (p-1)th term of the series.

Rather what we can do:
1. Start calculating Lucas-Lehmer series from 0th term and rather storing the whole term only store the s[i]%x (i.e. term modulo x).
2. Compute the next number of this modified series using the previous term. s[i] = (s[i-1]2 – 2)%x.
3. Compute up to (p-1)th term.
4. If the (p-1)th term is 0 then x is prime, otherwise not. Hence, s[p-1] has to be 0 to be x = (2p – 1) prime.

Examples:

Is 2^7 - 1 = 127 is a prime?
        so here x = 127, p = 7-1 = 6.
        Hence the modified Lucas-Lehmer series is:
        term 1: 4,
        term 2: (4*4 - 2) % 127 = 14,
        term 3: (14*14 - 2) % 127 = 67,
        term 4: (67*67 - 2) % 127 = 42,
        term 5: (42*42 - 2) % 127 = 111,
        term 6: (111*111) % 127 = 0.
        Here the 6th term is 0 so 127 is a prime number.

Code to check whether 2^p-1 is prime or not

C++

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// CPP program to check for primality using
// Lucas-Lehmer series.
#include <cmath>
#include <iostream>
using namespace std;
  
// Function to check whether (2^p - 1)
// is prime or not.
bool isPrime(int p) {
  
  // generate the number
  long long checkNumber = pow(2, p) - 1;
  
  // First number of the series
  long long nextval = 4 % checkNumber;
  
  // Generate the rest (p-2) terms
  // of the series.
  for (int i = 1; i < p - 1; i++) 
    nextval = (nextval * nextval - 2) % checkNumber;  
  
  // now if the (p-1)th term is
  // 0 return true else false.
  return (nextval == 0);
}
  
// Driver Program
int main() {
  // Check whether 2^p-1 is prime or not.
  int p = 7;
  
  long long checkNumber = pow(2, p) - 1;
  
  if (isPrime(p))
    cout << checkNumber << " is Prime.";
  else
    cout << checkNumber << " is not Prime.";
  
  return 0;
}

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Java

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// Java program to check for primality using
// Lucas-Lehmer series.
  
class GFG{
// Function to check whether (2^p - 1)
// is prime or not.
static boolean isPrime(int p) {
  
// generate the number
double checkNumber = Math.pow(2, p) - 1;
  
// First number of the series
double nextval = 4 % checkNumber;
  
// Generate the rest (p-2) terms
// of the series.
for (int i = 1; i < p - 1; i++) 
    nextval = (nextval * nextval - 2) % checkNumber; 
  
// now if the (p-1)th term is
// 0 return true else false.
return (nextval == 0);
}
  
// Driver Program
public static void main(String[] args) {
// Check whether 2^p-1 is prime or not.
int p = 7;
double checkNumber = Math.pow(2, p) - 1;
  
if (isPrime(p))
    System.out.println((int)checkNumber+" is Prime.");
else
    System.out.println((int)checkNumber+" is not Prime.");
  
}
}
// This code is contributed by mits

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Python3

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# Python3 Program to check for primality 
# using Lucas-Lehmer series.
  
# Function to check whether (2^p - 1)
# is prime or not.
def isPrime(p):
  
    # generate the number
    checkNumber = 2 ** p - 1
  
    # First number of the series
    nextval = 4 % checkNumber
  
    # Generate the rest (p-2) terms
    # of the series
    for i in range(1, p - 1):
        nextval = (nextval * nextval - 2) % checkNumber
  
    # now if the (p-1) the term is
    # 0 return true else false.
    if (nextval == 0): return True
    else: return False
  
# Driver Code
  
# Check whetherr 2^(p-1)
# is prime or not.
p = 7
checkNumber = 2 ** p - 1
  
if isPrime(p):
    print(checkNumber, 'is Prime.')
else:
    print(checkNumber, 'is not Prime')
  
# This code is contributed by egoista.

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C#

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// C# program to check for primality using
// Lucas-Lehmer series.
using System;
  
class GFG{
// Function to check whether (2^p - 1)
// is prime or not.
static bool isPrime(int p) {
  
// generate the number
double checkNumber = Math.Pow(2, p) - 1;
  
// First number of the series
double nextval = 4 % checkNumber;
  
// Generate the rest (p-2) terms
// of the series.
for (int i = 1; i < p - 1; i++) 
    nextval = (nextval * nextval - 2) % checkNumber; 
  
// now if the (p-1)th term is
// 0 return true else false.
return (nextval == 0);
}
  
// Driver Program
static void Main() {
// Check whether 2^p-1 is prime or not.
int p = 7;
double checkNumber = Math.Pow(2, p) - 1;
  
if (isPrime(p))
    Console.WriteLine((int)checkNumber+" is Prime.");
else
    Console.WriteLine((int)checkNumber+" is not Prime.");
  
}
}
// This code is contributed by mits

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PHP

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<?php
// PHP program to check for 
// primality using Lucas-
// Lehmer series.
  
// Function to check whether
/// (2^p - 1) is prime or not.
function isPrime($p
{
  
    // generate the number
        $checkNumber = pow(2, $p) - 1;
      
    // First number of the series
        $nextval = 4 % $checkNumber;
      
    // Generate the rest (p-2) terms
    // of the series.
    for ($i = 1; $i < $p - 1; $i++) 
        $nextval = ($nextval * $nextval - 2) % 
                                 $checkNumber
      
    // now if the (p-1)th term is
    // 0 return true else false.
    return ($nextval == 0);
}
  
    // Driver Code
    // Check whether 2^p-1 is
    // prime or not.
    $p = 7;
    $checkNumber = pow(2, $p) - 1;
    if (isPrime($p))
        echo $checkNumber , " is Prime.";
    else
        echo $checkNumber , " is not Prime.";
  
// This code is contributed by ajit.
?>

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Output:

127 is Prime.

The largest prime number at the time of writing this article is (2^(77232917) – 1) (discovered 2017-12-26). It has 23, 249, 425 digits. This prime numbers are found in the same way discussed above. Huge computational power and several months of processing is required to find out this kind of large prime numbers.
Interesting fact is that for checking this much big prime numbers, p are also taken prime. After processing if it finds that the number x is not prime then p is taken as the next prime number and the same process is run.



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I am an undergrad at IIEST Shibpur love to code and solve algorithm data structure problems

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