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.
C++
#include <iostream>
using namespace std;
int gcd( int a, int b)
{
if (a < b)
return gcd(b, a);
if (a % b == 0)
return b;
return gcd(b, a % b);
}
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;
}
bool isCarmichaelNumber( int n)
{
for ( int b = 2; b < n; b++) {
if (gcd(b, n) == 1)
if (power(b, n - 1, n) != 1)
return false ;
}
return true ;
}
int main()
{
cout << isCarmichaelNumber(500) << endl;
cout << isCarmichaelNumber(561) << endl;
cout << isCarmichaelNumber(1105) << endl;
return 0;
}
|
Java
import java.io.*;
class GFG {
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);
}
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;
}
static int isCarmichaelNumber( int n)
{
for ( int b = 2 ; b < n; b++) {
if (gcd(b, n) == 1 )
if (power(b, n - 1 , n) != 1 )
return 0 ;
}
return 1 ;
}
public static void main(String args[])
{
System.out.println(isCarmichaelNumber( 500 ));
System.out.println(isCarmichaelNumber( 561 ));
System.out.println(isCarmichaelNumber( 1105 ));
}
}
|
Python3
def gcd( a, b) :
if (a < b) :
return gcd(b, a)
if (a % b = = 0 ) :
return b
return gcd(b, a % b)
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
def isCarmichaelNumber( n) :
b = 2
while b<n :
if (gcd(b, n) = = 1 ) :
if (power(b, n - 1 , n) ! = 1 ):
return 0
b = b + 1
return 1
print (isCarmichaelNumber( 500 ))
print (isCarmichaelNumber( 561 ))
print (isCarmichaelNumber( 1105 ))
|
C#
using System;
class GFG {
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);
}
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;
}
static int isCarmichaelNumber( int n)
{
for ( int b = 2; b < n; b++) {
if (gcd(b, n) == 1)
if (power(b, n - 1, n) != 1)
return 0;
}
return 1;
}
public static void Main()
{
Console.WriteLine(isCarmichaelNumber(500));
Console.WriteLine(isCarmichaelNumber(561));
Console.WriteLine(isCarmichaelNumber(1105));
}
}
|
PHP
<?php
function gcd( $a , $b )
{
if ( $a < $b )
return gcd( $b , $a );
if ( $a % $b == 0)
return $b ;
return gcd( $b , $a % $b );
}
function 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 ;
}
function isCarmichaelNumber( $n )
{
for ( $b = 2; $b <= $n ; $b ++)
{
if (gcd( $b , $n ) == 1)
if (power( $b , $n - 1, $n ) != 1)
return 0;
}
return 1;
}
echo isCarmichaelNumber(500), " \n" ;
echo isCarmichaelNumber(561), "\n" ;
echo isCarmichaelNumber(1105), "\n" ;
?>
|
Javascript
<script>
function gcd(a, b)
{
if (a < b)
return gcd(b, a);
if (a % b == 0)
return b;
return gcd(b, a % b);
}
function power(x, y, mod)
{
if (y == 0)
return 1;
let temp = power(x, parseInt(y / 2, 10), mod) % mod;
temp = (temp * temp) % mod;
if (y % 2 == 1)
temp = (temp * x) % mod;
return temp;
}
function isCarmichaelNumber(n)
{
for (let b = 2; b < n; b++) {
if (gcd(b, n) == 1)
if (power(b, n - 1, n) != 1)
return 0;
}
return 1;
}
document.write(isCarmichaelNumber(500) + "</br>" );
document.write(isCarmichaelNumber(561) + "</br>" );
document.write(isCarmichaelNumber(1105));
</script>
|
C
#include<stdio.h>
int gcd( int a, int b)
{
if (a<b)
return gcd(b, a);
if (a % b == 0)
return b;
return gcd(b, a % b);
}
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;
}
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;
}
|
Output:
0
1
1
Time Complexity: O(n log n)
Auxiliary Space: O(n)
Last Updated :
08 May, 2023
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