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
p = 5 (p-1)! = 24 24 % 5 = 4 p = 7 (p-1)! = 6! = 720 720 % 7 = 6
How does it work?
1) We can quickly check result for p = 2 or p = 3.
2) 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).
3) 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)
How can it be useful?
Consider the problem of computing factorial under modulo of a prime number which is close to input number, i.e., we want to find value of “n! % p” such that n < p, p is a prime and n is close to p. For example (25! % 29). From Wilson’s theorem, we know that 28! is -1. So we basically need to find [ (-1) * inverse(28, 29) * inverse(27, 29) * inverse(26) ] % 29. The inverse function inverse(x, p) returns inverse of x under modulo p.
Disclaimer: This does not belong to TechCodeBit, its an article taken from the below
source and credits.
source and credits:http://www.geeksforgeeks.org/wilsons-theorem/
We have built the accelerating growth-oriented website for budding engineers and aspiring job holders of technology companies such as Google, Facebook, and Amazon
If you would like to study our free courses you can join us at
http://www.techcodebit.com. #techcodebit #google #microsoft #facebook #interview portal #jobplacements