Find the number of primitive roots modulo prime

Last Updated : 29 Jun, 2022

Given a prime p . The task is to count all the primitive roots of p .
A primitive root is an integer x (1 <= x < p) such that none of the integers x - 1, x² - 1, ...., x^{p - 2} - 1 are divisible by p but x^{p - 1} - 1 is divisible by p .
Examples:

Input: P = 3
Output: 1
The only primitive root modulo 3 is 2.
Input: P = 5
Output: 2
Primitive roots modulo 5 are 2 and 3.

Approach: There is always at least one primitive root for all primes. So, using Eulers totient function we can say that f(p-1) is the required answer where f(n) is euler totient function.
Below is the implementation of the above approach:

C++

// CPP program to find the number of
// primitive roots modulo prime
#include <bits/stdc++.h>
using namespace std;

// Function to return the count of
// primitive roots modulo p
int countPrimitiveRoots(int p)
{
    int result = 1;
    for (int i = 2; i < p; i++)
        if (__gcd(i, p) == 1)
            result++;

    return result;
}

// Driver code
int main()
{
    int p = 5;

    cout << countPrimitiveRoots(p - 1);

    return 0;
}

Java

 // Java program to find the number of
// primitive roots modulo prime

import java.io.*;

class GFG {
 // Recursive function to return gcd of a and b 
    static int __gcd(int a, int b) 
    { 
        // Everything divides 0  
        if (a == 0) 
          return b; 
        if (b == 0) 
          return a; 
       
        // base case 
        if (a == b) 
            return a; 
       
        // a is greater 
        if (a > b) 
            return __gcd(a-b, b); 
        return __gcd(a, b-a); 
    } 

// Function to return the count of
// primitive roots modulo p
static int countPrimitiveRoots(int p)
{
    int result = 1;
    for (int i = 2; i < p; i++)
        if (__gcd(i, p) == 1)
            result++;

    return result;
}

// Driver code
    public static void main (String[] args) {
            int p = 5;

    System.out.println( countPrimitiveRoots(p - 1));
    }
}
// This code is contributed by anuj_67..

Python3

# Python 3 program to find the number 
# of primitive roots modulo prime
from math import gcd

# Function to return the count of
# primitive roots modulo p
def countPrimitiveRoots(p):
    result = 1
    for i in range(2, p, 1):
        if (gcd(i, p) == 1):
            result += 1

    return result

# Driver code
if __name__ == '__main__':
    p = 5

    print(countPrimitiveRoots(p - 1))

# This code is contributed by
# Surendra_Gangwar

// C# program to find the number of 
// primitive roots modulo prime 
  
using System;
  
class GFG { 
 // Recursive function to return gcd of a and b  
    static int __gcd(int a, int b)  
    {  
        // Everything divides 0   
        if (a == 0)  
          return b;  
        if (b == 0)  
          return a;  
         
        // base case  
        if (a == b)  
            return a;  
         
        // a is greater  
        if (a > b)  
            return __gcd(a-b, b);  
        return __gcd(a, b-a);  
    }  
  
// Function to return the count of 
// primitive roots modulo p 
static int countPrimitiveRoots(int p) 
{ 
    int result = 1; 
    for (int i = 2; i < p; i++) 
        if (__gcd(i, p) == 1) 
            result++; 
  
    return result; 
} 
  
// Driver code 
     static public void Main (String []args) { 
            int p = 5; 
  
    Console.WriteLine( countPrimitiveRoots(p - 1)); 
    } 
} 
// This code is contributed by Arnab Kundu

PHP

<?php
// PHP program to find the number of
// primitive roots modulo prime

// Recursive function to return
// gcd of a and b 
function __gcd($a, $b) 
{ 
    // Everything divides 0 
    if ($a == 0) 
    return b; 
    
    if ($b == 0) 
    return $a; 
    
    // base case 
    if ($a == $b) 
        return $a; 
    
    // a is greater 
    if ($a > $b) 
        return __gcd($a - $b, $b); 
    return __gcd($a, $b - $a); 
} 

// Function to return the count of
// primitive roots modulo p
function countPrimitiveRoots($p)
{
    $result = 1;
    for ($i = 2; $i < $p; $i++)
        if (__gcd($i, $p) == 1)
            $result++;

    return $result;
}

// Driver code
$p = 5;

echo countPrimitiveRoots($p - 1);

// This code is contributed by anuj_67
?>

JavaScript

<script>

// Javascript program to find the number of
// primitive roots modulo prime

 // Recursive function to return gcd of a and b  
 function __gcd( a,  b)  
 {  
     // Everything divides 0   
     if (a == 0)  
       return b;  
     if (b == 0)  
       return a;  
      
     // base case  
     if (a == b)  
         return a;  
      
     // a is greater  
     if (a > b)  
         return __gcd(a-b, b);  
     return __gcd(a, b-a);  
 }  
  

// Function to return the count of
// primitive roots modulo p
function countPrimitiveRoots(p)
{
    var result = 1;
    for (var i = 2; i < p; i++)
        if (__gcd(i, p) == 1)
            result++;

    return result;
}

// Driver code
var p = 5;
document.write( countPrimitiveRoots(p - 1));

</script>

Output:

Time Complexity: O(p * log(min(a, b))), where a and b are two parameters of gcd.

Auxiliary Space: O(log(min(a, b)))

Compute power of power k times % m

pawan_asipu

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Find the number of primitive roots modulo prime

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