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 functionint 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 rangeimport 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 resultdef 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 codek = 3if 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 rangeusing 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
<?php// PHP program to find the // smallest twin in given range// Iterative Function to calculate// (a^n)%p in O(logy) function 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 & 1) $res = ($res * $a) % $p; // n must be even now $n = $n >> 1; // n = n/2 $a = ($a * $a) % $p; } return $res;}// If n is prime, then always // returns true, If n is// composite than returns // false with high probability// Higher value of k increases// probability of correct// result.function isPrime($n, $k){ // Corner cases if ($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 $a = 2 + rand() % ($n - 4); // Fermat's little theorem if (power($a, $n-1, $n) != 1) return false; $k--; } return true;}// Driver Code$k = 3;$res = isPrime(11, $k) ? " true\n": " false\n";echo($res);$res = isPrime(15, $k) ? " true\n": " false\n";echo($res);// This code is contributed by Ajit.?> |
Javascript
<script>// Javascript program to find the// smallest twin in given range /* Iterative Function to calculate// (a^n)%p in O(logy) */function power( a, n, p){ // Initialize result let 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.function isPrime( n, k){// Corner casesif (n <= 1 || n == 4) return false;if (n <= 3) return true; // Try k timeswhile (k > 0){ // Pick a random number in [2..n-2] // Above corner cases make sure that n > 4 let a = Math.floor(Math.random()* (n-1 - 2) + 2); // Fermat's little theorem if (power(a, n - 1, n) != 1) return false; k--; } return true;}// Driver Codelet k = 3;if(isPrime(11, k)) document.write(" true" + "</br>");else document.write(" false"+ "</br>");if(isPrime(15, k)) document.write(" true"+ "</br>");else document.write(" false"+ "</br>");</script> |
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
http://www.cse.iitk.ac.in/users/manindra/presentations/FLTBasedTests.pdf
https://en.wikipedia.org/wiki/Primality_test
This article is contributed by Ajay. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
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