Wilson's Theorem is a fundamental result in number theory that provides a necessary and sufficient condition for determining whether a given number is prime. It states that a natural number p > 1 is a prime number if and only if:
(p - 1)! ≡ −1 (mod p)
This means that the factorial of p - 1 (the product of all integers from 1 to p - 1) is congruent to −1 modulo p. In simpler terms, if you take the factorial of one less than a prime number and divide it by that prime number, the remainder will be p - 1.
Wilson's Theorem Examples
Example for Prime Number
Let’s check for p = 5
Calculate (5 - 1)! = 4! = 24
Check 24 mod 5 = 4 which is indeed -1 mod 5
Thus, 5 is prime.
Example of Composite Number:
For p = 4
Calculate (4 - 1)! = 3! = 6
Check 6 mod 4 = 2 which is not -1 mod 4
Thus, 4 is not prime.
Applications of Wilson's Theorem
While Wilson's Theorem is theoretically significant, its direct application in practical prime number testing is limited due to computational inefficiency. Nonetheless, it serves as a fundamental example of prime number properties in theoretical mathematics and is often referenced in discussions about primality testing.
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