Carmichael Numbers (original) (raw)

Last Updated : 8 May, 2023

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.
Examples :

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.

Try It Yourself

C++ `

// A C++ program to check if a number is // Carmichael or not. #include 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.

Python3

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.

C#

// C# program to check if a number is // Carmichael or not. using System;

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()
{
    Console.WriteLine(isCarmichaelNumber(500));
    Console.WriteLine(isCarmichaelNumber(561));
    Console.WriteLine(isCarmichaelNumber(1105));
}

}

// This code is contributed by vt_m.

PHP

JavaScript

C

// C Program to find if a number is Carmichael Number #include<stdio.h> int gcd(int a, int b) //Function to find GCD { if (a<b) return gcd(b, a); if (a % b == 0) return b; return gcd(b, a % b); }

// 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; }

//Function to find if received number n is a Carmichael number int carmichaelnumber(int n)
{ for (int b=2;b<n;b++) { if (gcd(b,n)==1) if (power(b,n-1,n)!= 1) { printf("0"); return 0; } } printf("1"); return 0; }; int main() { carmichaelnumber(500); printf("\n"); carmichaelnumber(561); printf("\n"); carmichaelnumber(1105); return 0;

// This code is contributed by Susobhan Akhuli

}

`

Output:

0 1 1

Time Complexity: O(n log n)
Auxiliary Space: O(n)