Implementation of Wilson Primality test - GeeksforGeeks

09 Apr, 2021
Given a number N, the task is to check if it is prime or not using Wilson Primality Test. Print ‘1’ isf 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
(p - 1)! = 24
24 % 5  = 4

Input: p = 7
Output: Yes
(p-1)! = 6! = 720
720 % 7  = 6
Below is the implementation of Wilson Primality Test  
C++
Java
Python3
C#
Javascript
// 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 implementation to check if a number is // prime or not using Wilson Primality Testpublic 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 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# implementation to check if a number is // prime or not using Wilson Primality Testusing 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
<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>
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
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)
 
divyesh072019
divyeshrabadiya07
rag2127
suresh07
factorial
Modular Arithmetic
Prime Number
Algorithms
Mathematical
Mathematical
Prime Number
Modular Arithmetic
factorial
Algorithms
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
SDE SHEET - A Complete Guide for SDE Preparation
DSA Sheet by Love Babbar
How to write a Pseudo Code?
Understanding Time Complexity with Simple Examples
Introduction to Algorithms
Program for Fibonacci numbers
Write a program to print all permutations of a given string
C++ Data Types
Set in C++ Standard Template Library (STL)
Coin Change | DP-7