wilson定理（Wilson Theorem An Insight into Prime Numbers）

Wilson Theorem is one of the most significant discoveries in number theory that provides insight into the behavior of prime numbers. In this article, we will delve into the details of Wilson Theorem, its proof and significance.

What is Wilson Theorem?

Wilson Theorem states that a positive integer p greater than 1 is a prime number if and only if the value (p - 1)! + 1 is divisible by p.

For example, consider the prime number p = 5. According to Wilson Theorem, (5 - 1)! + 1 = 24 + 1 = 25 which is divisible by 5. Similarly, for a composite number such as 4, (4 - 1)! + 1 = 7 which is not divisible by 4.

Proof of Wilson Theorem

Wilson Theorem can be proved using the concept of Fermat's little theorem which states that if p is a prime number, then for any integer a, a raised to the power of p modulo p is congruent to a modulo p. Mathematically, a^p ≡ a (mod p).

Now, let us assume that p is a prime number. Then, we have (p-1)! ≡ -1 (mod p) from the equation (p-1)!+1 ≡ 0 (mod p). From this, we can say that the product of all numbers in the set S={1, 2, ..., p-1} is congruent to -1 modulo p.

Now, let us consider the set T={1, 2, ..., p-1}. We can pair every element of this set with its inverse modulo p. For example, for p=7, T={1, 2, 3, 4, 5, 6} and the pairs are (1, 1), (2, 4), (3, 5), (6, 6).

Since p is a prime number, every element in T has a unique inverse modulo p. The only exception is the element 1, which is its own inverse. Now, we can divide T into two sets: {1} and U, where U is the set of all elements except for 1. The product of elements in U can be written as follows:

1 × (2 × 4) × (3 × 5) × ... × (p-2 × (p-4)) × (p-1) ≡ -1 (mod p)

Since every element has an inverse in modulo p, we can easily calculate the product of all elements in U. Using Fermat's little theorem, we can show that a × (p-1-a) ≡ -1 (mod p) for every a in U. Therefore, the product of all elements in U can be simplified as (-1)^u, where u is the size of the set U.

Since the product of all elements in S is congruent to -1 modulo p and the product of all elements in U is congruent to (-1)^u, we can say that (p-1)! ≡ (-1)^u (mod p). Now, if p divides (p-1)!, then it must also divide (-1)^u. Therefore, p is a prime number if and only if (-1)^u ≡ -1 (mod p), which is equivalent to u being even.

Significance of Wilson Theorem

Wilson Theorem is a powerful tool for determining whether a given number is prime or composite. However, it is not an efficient algorithm for large numbers. This is because computing the factorial of a large integer is a time-consuming process.

Nevertheless, Wilson Theorem has several interesting applications in number theory and cryptography. It is used in the generation of pseudorandom numbers and in the design of cryptographic protocols. Additionally, Wilson primes, which are prime numbers that satisfy the equation (p-1)! ≡ -1 (mod p^2), have been extensively studied by mathematicians and are an important area of research in number theory.

In conclusion, Wilson Theorem provides a unique insight into prime numbers and their properties. It has several applications in number theory and cryptography and is an important area of research in mathematics.