Carmichael Numbers

2.5

A number n is said to be a Carmichael number if it satisfies the following modular arithmetic condition:

  power(b, n-1) MOD n = 1, 
  for all b ranging from 1 to n such that b and 
  n are relatively prime, i.e, gcd(b, n) = 1 

Given a positive integer n, find if it is a Carmichael number. These numbers have importance in Fermat Method for primality testing.

Input :  n = 8
Output : false
Explanation : 8 is not a Carmichael number because 3 is 
              relatively prime to 8 and (38-1) % 8
              = 2187 % 8 is not 1.
              
Input :  n = 561
Output : true

The idea is simple, we iterate through all numbers from 1 to n and for every relatively prime number, we check if its (n-1)th power under modulo n is 1 or not.

Below is a the program to check if a given number is Carmichael or not.

C++

// A C++ program to check if a number is
// Carmichael or not.
#include <iostream>
using namespace std;

// utility function to find gcd of two numbers
int gcd(int a,int b)
{
    if (a < b)
        return gcd(b, a);
    if (a%b == 0)
        return b;
    return gcd(b, a%b);
}

// utility function to find pow(x, y) under
// given modulo mod
int power(int x, int y, int mod)
{
    if (y==0)
        return 1;
    int temp = power(x, y/2, mod) % mod;
    temp = (temp*temp) % mod;
    if (y%2==1)
        temp = (temp*x) % mod;
    return temp;
}

// This function receives an integer n and
// finds if it's a Carmichael number
bool isCarmichaelNumber(int n)
{
    for (int b=2; b<n; b++)
    {
        // If "b" is relatively prime to n
        if (gcd(b, n) == 1)

            // And pow(b, n-1)%n is not 1,
            // return false.
            if (power(b, n-1, n) != 1)
                return false;
    }
    return true;
}

// Driver function
int main()
{
    cout << isCarmichaelNumber(500) << endl;
    cout << isCarmichaelNumber(561) << endl;
    cout << isCarmichaelNumber(1105) << endl;
    return 0;
}

Java

// JAVA program to check if a number is
// Carmichael or not.
import java.io.*;

class GFG{

    // utility function to find gcd of
    // two numbers
    static int gcd(int a,int b)
    {
        if (a < b)
            return gcd(b, a);
        if (a % b == 0)
            return b;
        return gcd(b, a % b);
    }
    
    // utility function to find pow(x, y) 
    // under given modulo mod
    static int power(int x, int y, int mod)
    {
        if (y == 0)
            return 1;
        int temp = power(x, y / 2, mod) % mod;
        temp = (temp * temp) % mod;
        if (y % 2 == 1)
            temp = (temp*x) % mod;
        return temp;
    }

    // This function receives an integer n and
    // finds if it's a Carmichael number
    static int isCarmichaelNumber(int n)
    {
        for (int b = 2; b < n; b++)
        {
            // If "b" is relatively prime to n
            if (gcd(b, n) == 1)

                // And pow(b, n-1)%n is not 1,
                // return false.
                if (power(b, n - 1, n) != 1)
                    return 0;
        }
        return 1;
    }

    // Driver function
    public static void main(String args[])
    {
        System.out.println(isCarmichaelNumber(500)); 
        System.out.println(isCarmichaelNumber(561));
        System.out.println(isCarmichaelNumber(1105)); 
    }
}
// This code is contributed by Nikita Tiwari. 

Python

# A Python program to check if a number is
# Carmichael or not.

# utility function to find gcd of two numbers
def gcd( a,b) :
    if (a < b) :
        return gcd(b, a)
    if (a % b == 0) :
        return b
    return gcd(b, a % b)

# utility function to find pow(x, y) under
# given modulo mod
def power(x, y, mod) :
    if (y == 0) :
        return 1
    temp = power(x, y / 2, mod) % mod
    temp = (temp * temp) % mod
    if (y % 2 == 1) :
        temp = (temp * x) % mod
    return temp


# This function receives an integer n and
# finds if it's a Carmichael number
def isCarmichaelNumber( n) :
    b = 2
    while b<n :
        
        # If "b" is relatively prime to n
        if (gcd(b, n) == 1) :

            # And pow(b, n-1)%n is not 1,
            # return false.
            if (power(b, n - 1, n) != 1):
                return 0
        b = b + 1
    return 1

# Driver function
print isCarmichaelNumber(500)
print isCarmichaelNumber(561)
print isCarmichaelNumber(1105) 

#This code is contributed by Nikita Tiwari.


Output:

0
1
1

This article is contributed by Ashutosh Kumar. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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