Given a number N and a base number A. The task is to check whether the number is a Fermat Pseudoprime to the base.
The number N is called as Fermat Pseudoprime to the base A, if
1. A > 1
2. N is a composite number
3. N divides AN-1 – 1.
Examples:
Input : N = 645, a = 2
Output :1
645 = 3*5*43, Hence it is a composite number
Also 645 divides 2^(644)-1
Hence it is a Fermat Pseudoprime.
Input : N = 6, a = 2
Output :0
Approach: The approach is to check the below conditions:
- Check if A > 1.
- Check if N is a composite number.
- Check if N divides AN-1 – 1.
If all of the above conditions satisfy then N is a fermat pseudoprime to base A.
Below is the implementation of the above approach:
C++
// C++ program to check if N is Fermat pseudoprime// to the base A or not#include <bits/stdc++.h>using namespace std;// Function to check if the given number is compositebool checkcomposite(int n){ // Check if there is any divisor of n less than sqrt(n) for (int i = 2; i <= sqrt(n); i++) { if (n % i == 0) return 1; } return 0;}// Effectively calculate (x^y) modulo modint power(int x, int y, int mod){ // Initialize result int res = 1; while (y) { // If power is odd, then update the answer if (y & 1) res = (res * x) % mod; // Square the number and reduce // the power to its half y = y >> 1; x = (x * x) % mod; } // Return the result return res;}// Function to check for Fermat Pseudoprimebool Check(int n, int a){ // If it is composite and satisfy Fermat criterion if (a>1 && checkcomposite(n) && power(a, n - 1, n) == 1) return 1; // Else return 0 return 0;}// Driver codeint main(){ int N = 645; int a = 2; // Function call cout << Check(N, a); return 0;} |
Java
// Java program to check if N is Fermat pseudoprime // to the base A or not class GFG { // Function to check if // the given number is composite static boolean checkcomposite(int n) { // Check if there is any divisor of n // less than sqrt(n) for (int i = 2; i <= Math.sqrt(n); i++) { if (n % i == 0) { return true; } } return false; } // Effectively calculate (x^y) modulo mod static int power(int x, int y, int mod) { // Initialize result int res = 1; while (y != 0) { // If power is odd, // then update the answer if ((y & 1) == 1) { res = (res * x) % mod; } // Square the number and reduce // the power to its half y = y >> 1; x = (x * x) % mod; } // Return the result return res; } // Function to check for Fermat Pseudoprime static int Check(int n, int a) { // If it is composite and // satisfy Fermat criterion if (a > 1 && checkcomposite(n) && power(a, n - 1, n) == 1) { return 1; } // Else return 0 return 0; } // Driver Code public static void main(String[] args) { int N = 645; int a = 2; // Function call System.out.println(Check(N, a)); }}// This code is contributed by 29AjayKumar |
Python3
# Python3 program to check if N is Fermat pseudoprime# to the base A or notfrom math import sqrt# Function to check if the given number is compositedef checkcomposite(n): # Check if there is any divisor of n less than sqrt(n) for i in range(2,int(sqrt(n))+1,1): if (n % i == 0): return 1 return 0# Effectively calculate (x^y) modulo moddef power(x, y, mod): # Initialize result res = 1 while (y): # If power is odd, then update the answer if (y & 1): res = (res * x) % mod # Square the number and reduce # the power to its half y = y >> 1 x = (x * x) % mod # Return the result return res# Function to check for Fermat Pseudoprimedef Check(n,a): # If it is composite and satisfy Fermat criterion if (a>1 and checkcomposite(n) and power(a, n - 1, n) == 1): return 1 # Else return 0 return 0# Driver codeif __name__ == '__main__': N = 645 a = 2 # Function call print(Check(N, a))# This code is contributed by# Surendra_Gangwar |
C#
// C# program to check if N is Fermat pseudoprime // to the base A or not using System;class GFG{ // Function to check if // the given number is composite static bool checkcomposite(int n) { // Check if there is any divisor of n // less than sqrt(n) for (int i = 2; i <= Math.Sqrt(n); i++) { if (n % i == 0) return true; } return false; } // Effectively calculate (x^y) modulo mod static int power(int x, int y, int mod) { // Initialize result int res = 1; while (y != 0) { // If power is odd, then update the answer if ((y & 1) == 1) res = (res * x) % mod; // Square the number and reduce // the power to its half y = y >> 1; x = (x * x) % mod; } // Return the result return res; } // Function to check for Fermat Pseudoprime static int Check(int n, int a) { // If it is composite and satisfy Fermat criterion if (a > 1 && checkcomposite(n) && power(a, n - 1, n) == 1) return 1; // Else return 0 return 0; } // Driver code static public void Main () { int N = 645; int a = 2; // Function call Console.WriteLine(Check(N, a)); }}// This code is contributed by AnkitRai01 |
Javascript
<script>// Javascript program to check if// N is Fermat pseudoprime// to the base A or not// Function to check if the given// number is compositefunction checkcomposite(n){ // Check if there is any divisor // of n less than sqrt(n) for (let i = 2; i <= Math.sqrt(n); i++) { if (n % i == 0) return 1; } return 0;}// Effectively calculate (x^y) modulo modfunction power(x, y, mod){ // Initialize result let res = 1; while (y) { // If power is odd, then update the answer if (y & 1) res = (res * x) % mod; // Square the number and reduce // the power to its half y = y >> 1; x = (x * x) % mod; } // Return the result return res;}// Function to check for Fermat Pseudoprimefunction Check(n, a){ // If it is composite and satisfy // Fermat criterion if (a>1 && checkcomposite(n) && power(a, n - 1, n) == 1) return 1; // Else return 0 return 0;}// Driver code let N = 645; let a = 2; // Function call document.write(Check(N, a));</script> |
Output:
1
Time Complexity : O(sqrt(N))
Auxiliary Space: O(1)
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