Lehmann’s Primality Test

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.