Fermat Method of Primality Test (original) (raw)

Last Updated : 23 Jul, 2025

Given a number n, check if it is prime or not. We have introduced and discussed the School method for primality testing in Set 1.
Introduction to Primality Test and School Method
In this post, Fermat's method is discussed. This method is a probabilistic method and is based on Fermat's Little Theorem.

Fermat's Little Theorem: If n is a prime number, then for every a, 1 < a < n-1,

an-1 ? 1 (mod n) OR an-1 % n = 1

Example: Since 5 is prime, 24 ? 1 (mod 5) [or 24%5 = 1], 34 ? 1 (mod 5) and 44 ? 1 (mod 5)

     Since 7 is prime, 26 ? 1 (mod 7),
     36 ? 1 (mod 7), 46 ? 1 (mod 7) 
     56 ? 1 (mod 7) and 66 ? 1 (mod 7) 

Refer this for different proofs.

If a given number is prime, then this method always returns true. If the given number is composite (or non-prime), then it may return true or false, but the probability of producing incorrect results for composite is low and can be reduced by doing more iterations.

Below is algorithm:

// Higher value of k indicates probability of correct // results for composite inputs become higher. For prime // inputs, result is always correct

1. Repeat following k times: a) Pick a randomly in the range [2, n - 2] b) If gcd(a, n) ? 1, then return false c) If an-1 ≢ 1 (mod n), then return false
2. Return true [probably prime].

Below is the implementation of the above algorithm. The code uses power function from Modular Exponentiation

C++ `

// C++ program to find the smallest twin in given range #include <bits/stdc++.h> using namespace std;

/* Iterative Function to calculate (a^n)%p in O(logy) */ int power(int a, unsigned int n, int p) { int res = 1; // Initialize result a = a % p; // Update 'a' if 'a' >= p

while (n > 0)
{
    // If n is odd, multiply 'a' with result
    if (n & 1)
        res = (res*a) % p;

    // n must be even now
    n = n>>1; // n = n/2
    a = (a*a) % p;
}
return res;

}

/Recursive function to calculate gcd of 2 numbers/ int gcd(int a, int b) { if(a < b) return gcd(b, a); else if(a%b == 0) return b; else return gcd(b, a%b);
}

// If n is prime, then always returns true, If n is // composite than returns false with high probability // Higher value of k increases probability of correct // result. bool isPrime(unsigned int n, int k) { // Corner cases if (n <= 1 || n == 4) return false; if (n <= 3) return true;

// Try k times while (k>0) { // Pick a random number in [2..n-2]
// Above corner cases make sure that n > 4 int a = 2 + rand()%(n-4);

   // Checking if a and n are co-prime
   if (gcd(n, a) != 1)
      return false;

   // Fermat's little theorem
   if (power(a, n-1, n) != 1)
      return false;

   k--;
}

return true;

}

// Driver Program to test above function int main() { int k = 3; isPrime(11, k)? cout << " true\n": cout << " false\n"; isPrime(15, k)? cout << " true\n": cout << " false\n"; return 0; }

Java

// Java program to find the // smallest twin in given range

import java.io.; import java.math.;

class GFG {

/* Iterative Function to calculate
// (a^n)%p in O(logy) */
static int power(int a,int n, int p)
{
    // Initialize result
    int res = 1;
    
    // Update 'a' if 'a' >= p
    a = a % p; 

    while (n > 0)
    {
        // If n is odd, multiply 'a' with result
        if ((n & 1) == 1)
            res = (res * a) % p;

        // n must be even now
        n = n >> 1; // n = n/2
        a = (a * a) % p;
    }
    return res;
}

// If n is prime, then always returns true, 
// If n is composite than returns false with 
// high probability Higher value of k increases
//  probability of correct result.
static boolean isPrime(int n, int k)
{
// Corner cases
if (n <= 1 || n == 4) return false;
if (n <= 3) return true;

// Try k times
while (k > 0)
{
    // Pick a random number in [2..n-2]     
    // Above corner cases make sure that n > 4
    int a = 2 + (int)(Math.random() % (n - 4)); 

    // Fermat's little theorem
    if (power(a, n - 1, n) != 1)
        return false;

    k--;
    }

    return true;
}

// Driver Program 
public static void main(String args[])
{
    int k = 3;
    if(isPrime(11, k))
        System.out.println(" true");
    else
        System.out.println(" false");
    if(isPrime(15, k))
        System.out.println(" true");
    else
        System.out.println(" false");
        
}

}

// This code is contributed by Nikita Tiwari.

Python3

Python3 program to find the smallest

twin in given range

import random

Iterative Function to calculate

(a^n)%p in O(logy)

def power(a, n, p):

# Initialize result 
res = 1 

# Update 'a' if 'a' >= p 
a = a % p  

while n > 0:
    
    # If n is odd, multiply 
    # 'a' with result 
    if n % 2:
        res = (res * a) % p
        n = n - 1
    else:
        a = (a ** 2) % p
        
        # n must be even now 
        n = n // 2
        
return res % p

If n is prime, then always returns true,

If n is composite than returns false with

high probability Higher value of k increases

probability of correct result

def isPrime(n, k):

# Corner cases
if n == 1 or n == 4:
    return False
elif n == 2 or n == 3:
    return True

# Try k times 
else:
    for i in range(k):
        
        # Pick a random number 
        # in [2..n-2]      
        # Above corner cases make 
        # sure that n > 4 
        a = random.randint(2, n - 2)
        
        # Fermat's little theorem 
        if power(a, n - 1, n) != 1:
            return False
            
return True
        

Driver code

k = 3 if isPrime(11, k): print("true") else: print("false")

if isPrime(15, k): print("true") else: print("false")

This code is contributed by Aanchal Tiwari

C#

// C# program to find the // smallest twin in given range using System; class GFG {

/* Iterative Function to calculate
// (a^n)%p in O(logy) */
static int power(int a,int n, int p)
{
    // Initialize result
    int res = 1;
     
    // Update 'a' if 'a' >= p
    a = a % p; 
 
    while (n > 0)
    {
        // If n is odd, multiply 'a' with result
        if ((n & 1) == 1)
            res = (res * a) % p;
 
        // n must be even now
        n = n >> 1; // n = n/2
        a = (a * a) % p;
    }
    return res;
}
 
// If n is prime, then always returns true, 
// If n is composite than returns false with 
// high probability Higher value of k increases
//  probability of correct result.
static bool isPrime(int n, int k)
{
    // Corner cases
    if (n <= 1 || n == 4) return false;
    if (n <= 3) return true;
     
    // Try k times
    while (k > 0)
    {
        // Pick a random number in [2..n-2]     
        // Above corner cases make sure that n > 4
        Random rand = new Random(); 
        int a = 2 + (int)(rand.Next() % (n - 4)); 
     
        // Fermat's little theorem
        if (power(a, n - 1, n) != 1)
            return false;
     
        k--;
    }
 
    return true;
}

static void Main() { int k = 3; if(isPrime(11, k)) Console.WriteLine(" true"); else Console.WriteLine(" false"); if(isPrime(15, k)) Console.WriteLine(" true"); else Console.WriteLine(" false"); } }

// This code is contributed by divyesh072019

PHP

JavaScript

`

Output:

true false

Time complexity: O(k Log n). Note that the power function takes O(Log n) time.

Auxiliary Space: O(min(log a, log b))
Note that the above method may fail even if we increase the number of iterations (higher k). There exist some composite numbers with the property that for every a < n and gcd(a, n) = 1 we have an-1 ? 1 (mod n). Such numbers are called Carmichael numbers. Fermat's primality test is often used if a rapid method is needed for filtering, for example in the key generation phase of the RSA public key cryptographic algorithm.

We will soon be discussing more methods for Primality Testing.

References:
https://en.wikipedia.org/wiki/Fermat_primality_test
https://en.wikipedia.org/wiki/Prime_number
https://www.cse.iitk.ac.in/users/manindra/presentations/FLTBasedTests.pdf
https://en.wikipedia.org/wiki/Primality_test