Skip to content
Related Articles

Related Articles

Lehmann’s Primality Test

Improve Article
Save Article
  • Last Updated : 13 Oct, 2022
Improve Article
Save Article

An integer p greater than one is prime if the only divisors of p are 1 and p. First few prime numbers are 2, 3, 5, 7, 11, 13, …

The Lehmann’s test is a probabilistic primality test for an natural number n, it can test the primality of any kind of number(whether a large odd number is prime or not). The Lehmann’s test is a variation of Fermat’s Primality Test.
The approach used is as follows: 

If ‘n’ is an odd number and ‘a’ is a random integer less than n but greater than 1, then 

x = (a^((n-1)/2)) (mod n)

It is computed.  

  1. If x is 1 or -1(or n-1), then n may be prime. 
  2. If x is not 1 or -1(or n-1), then n is definitely composite. 

The fact that any composite number can be turned out to be a prime, in this case, depends on the random value ‘a’. If all the values of a and n are co-prime, then n can be said as a prime number. 

Example-1:
Input: n = 13 
Output: 13 is Prime 

Explanation: 
Let a = 3, then, 
3^((13-1)/2) % 13 = 729 % 13 = 1
Hence, 13 is Prime.

Example-2:
Input: n = 91
Output: 91 is Composite

Explanation: 
Let a = 3, then, 
3^((91-1)/2) % 91 = 27
Hence, 91 is Composite. 

C++




// C++ code for Lehmann's Primality Test
#include<stdio.h>
#include<stdlib.h>
#include<ctime>
#include<bits/stdc++.h>
using namespace std;
 
// function to check Lehmann's test
int lehmann(int n, int t)
{
 
    // generating a random base less than n
    int a = 2 + (rand() % (n - 1));
 
    // calculating exponent
    int e = (n - 1) / 2;
 
    // iterate to check for different base values
    // for given number of tries 't'
    while(t > 0)
    {
 
        // calculating final value using formula
        int result =((int)(pow(a, e)))% n;
 
        //if not equal, try for different base
        if((result % n) == 1 || (result % n) == (n - 1))
        {
            a = 2 + (rand() % (n - 1));
            t -= 1;
        }
 
        // else return negative
        else
            return -1;
    }
 
    // return positive after attempting
    return 1;
}
 
// Driver code
int main()
{
    int n = 13 ; // number to be tested
    int t = 10 ; // number of tries
 
    // if n is 2, it is prime
    if(n == 2)
    cout << "2 is Prime.";
 
    // if even, it is composite
    if(n % 2 == 0)
        cout << n << " is Composite";
 
    // if odd, check
    else
    {
        int flag = lehmann(n, t);
     
        if(flag ==1)
            cout << n << " may be Prime.";
     
        else
            cout << n << " is Composite.";
    }
}
 
// This code is contributed by chitranayal


Java




// Java code for Lehmann's Primality Test
     
// importing "random" for random operations
import java.util.Random;
 
class GFG
{
 
    // function to check Lehmann's test
    static int lehmann(int n, int t)
    {
     
        // create instance of Random class
        Random rand = new Random();
         
        // generating a random base less than n
        int a = rand.nextInt(n - 3) + 2;
     
        // calculating exponent
        float e = (n - 1) / 2;
     
        // iterate to check for different base values
        // for given number of tries 't'
        while(t > 0)
        {
     
            // calculating final value using formula
            int result = ((int)(Math.pow(a, e))) % n;
     
            // if not equal, try for different base
            if((result % n) == 1 || (result % n) == (n - 1))
            {
                a = rand.nextInt(n - 3) + 2;
                t -= 1;
            }
     
            // else return negative
            else
                return -1;
                 
        }
         
        // return positive after attempting
        return 1;
    }
     
    // Driver code
    public static void main (String[] args)
    {
    int n = 13; // number to be tested
    int t = 10; // number of tries
     
    // if n is 2, it is prime
    if(n == 2)
        System.out.println(" 2 is Prime.");
     
    // if even, it is composite
    if(n % 2 == 0)
        System.out.println(n + " is Composite");
     
    // if odd, check
    else
    {
        long flag = lehmann(n, t);
     
        if(flag == 1)
            System.out.println(n + " may be Prime.");
     
        else
            System.out.println(n + " is Composite.");
    }
}
}
 
// This code is contributed by AnkitRai01


Python3




# Python code for Lehmann's Primality Test
 
# importing "random" for random operations
import random
 
# function to check Lehmann's test
def lehmann(n, t):
 
    # generating a random base less than n
    a = random.randint(2, n-1)
 
    # calculating exponent
    e =(n-1)/2
 
    # iterate to check for different base values
    # for given number of tries 't'
    while(t>0):
 
        # calculating final value using formula
        result =((int)(a**e))% n
 
        # if not equal, try for different base
        if((result % n)== 1 or (result % n)==(n-1)):
            a = random.randint(2, n-1)
            t-= 1
 
        # else return negative
        else:
            return -1
 
    # return positive after attempting
    return 1
 
# Driver code
n = 13    # number to be tested
t = 10    # number of tries
 
# if n is 2, it is prime
if(n is 2):
    print("2 is Prime.")
 
# if even, it is composite
if(n % 2 == 0):
    print(n, "is Composite")
 
# if odd, check
else:
    flag = lehmann(n, t)
 
    if(flag is 1):
        print(n, "may be Prime.")
 
    else:
        print(n, "is Composite.")


C#




// C# code for Lehmann's Primality Test
using System;
 
class GFG
{
 
// function to check Lehmann's test
static int lehmann(int n, int t)
{
 
    // create instance of Random class
    Random rand = new Random();
     
    // generating a random base less than n
    int a = rand.Next(n - 3) + 2;
 
    // calculating exponent
    float e = (n - 1) / 2;
 
    // iterate to check for different base values
    // for given number of tries 't'
    while(t > 0)
    {
 
        // calculating final value using formula
        int result = ((int)(Math.Pow(a, e))) % n;
 
        // if not equal, try for different base
        if((result % n) == 1 ||
           (result % n) == (n - 1))
        {
            a = rand.Next(n - 3) + 2;
            t -= 1;
        }
 
        // else return negative
        else
            return -1;
             
    }
     
    // return positive after attempting
    return 1;
}
 
// Driver code
public static void Main (String[] args)
{
    int n = 13; // number to be tested
    int t = 10; // number of tries
     
    // if n is 2, it is prime
    if(n == 2)
        Console.WriteLine(" 2 is Prime.");
     
    // if even, it is composite
    if(n % 2 == 0)
        Console.WriteLine(n + " is Composite");
     
    // if odd, check
    else
    {
        long flag = lehmann(n, t);
     
        if(flag == 1)
            Console.WriteLine(n + " may be Prime.");
     
        else
            Console.WriteLine(n + " is Composite.");
    }
}
}
 
// This code is contributed by Rajput-Ji


Output: 

13 may be Prime.

 


My Personal Notes arrow_drop_up
Related Articles

Start Your Coding Journey Now!