Given a number N, the task is to check if it is prime or not using Wilson Primality Test. Print ‘1’ if the number is prime, else print ‘0’.
Wilson’s theorem states that a natural number p > 1 is a prime number if and only if
(p - 1) ! ? -1 mod p OR (p - 1) ! ? (p-1) mod p
Examples:
Input: p = 5
Output: Yes
Explanation: (p – 1)! = 24
24 % 5 = 4Input: p = 7
Output: Yes
Explanation: (p-1)! = 6! = 720
720 % 7 = 6
Below is the implementation of the Wilson Primality Test
C++
// C++ implementation to check if a number is // prime or not using Wilson Primality Test#include <bits/stdc++.h>using namespace std;// Function to calculate the factoriallong fact(const int& p){ if (p <= 1) return 1; return p * fact(p - 1);}// Function to check if the// number is prime or notbool isPrime(const int& p){ if (p == 4) return false; return bool(fact(p >> 1) % p);}// Driver codeint main(){ cout << isPrime(127); return 0;} |
Java
// Java implementation to check if a number is // prime or not using Wilson Primality Test public class Main{ // Function to calculate the factorial public static long fact(int p) { if (p <= 1) return 1; return p * fact(p - 1); } // Function to check if the // number is prime or not public static long isPrime(int p) { if (p == 4) return 0; return (fact(p >> 1) % p); } public static void main(String[] args) { if(isPrime(127) == 0) { System.out.println(0); } else{ System.out.println(1); } }}// This code is contributed by divyesh072019 |
Python3
# Python3 implementation to check if a number is # prime or not using Wilson Primality Test# Function to calculate the factorialdef fact(p): if (p <= 1): return 1 return p * fact(p - 1)# Function to check if the# number is prime or notdef isPrime(p): if (p == 4): return 0 return (fact(p >> 1) % p)# Driver codeif (isPrime(127) == 0): print(0)else: print(1)# This code is contributed by rag2127 |
C#
// C# implementation to check if a number is // prime or not using Wilson Primality Test using System;class GFG { // Function to calculate the factorial static long fact(int p) { if (p <= 1) return 1; return p * fact(p - 1); } // Function to check if the // number is prime or not static long isPrime(int p) { if (p == 4) return 0; return (fact(p >> 1) % p); } static void Main() { if(isPrime(127) == 0) { Console.WriteLine(0); } else{ Console.WriteLine(1); } }}// This code is contributed by divyeshrabadiya07 |
Javascript
<script> // Javascript implementation to check if a number is // prime or not using Wilson Primality Test // Function to calculate the factorial function fact(p) { if (p <= 1) return 1; return p * fact(p - 1); } // Function to check if the // number is prime or not function isPrime(p) { if (p == 4) return false; if(fact(p >> 1) % p == 0) { return 0; } return 1; } document.write(isPrime(127)); // This code is contributed by suresh07.</script> |
Output
1
How does it work?
- We can quickly check result for p = 2 or p = 3.
- For p > 3: If p is composite, then its positive divisors are among the integers 1, 2, 3, 4, … , p-1 and it is clear that gcd((p-1)!,p) > 1, so we can not have (p-1)! = -1 (mod p).
- Now let us see how it is exactly -1 when p is a prime. If p is a prime, then all numbers in [1, p-1] are relatively prime to p. And for every number x in range [2, p-2], there must exist a pair y such that (x*y)%p = 1. So
[1 * 2 * 3 * … (p-1)]%p
= [1 * 1 * 1 … (p-1)] // Group all x and y in [2..p-2] such that (x*y)%p = 1
= (p-1)
Time Complexity: O(N) as recursive factorial function takes O(N) time
Auxiliary Space: O(N), for using recursive stack space.
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