Wilson prime
Definition
A Wilson prime is a specific type of prime number, named after English mathematician John Wilson. It is a prime number p that satisfies the condition ((p-1)! + 1) mod p^2 = 0, where "!" denotes the factorial function. This condition is a derivative of Wilson's Theorem, which states that for any prime number p, (p-1)! + 1 is divisible by p. However, in the case of Wilson primes, the result of the factorial function plus one is not just divisible by p, but by p squared.
History
The concept of Wilson primes was first introduced by John Wilson, an English mathematician, in the 18th century. However, the term "Wilson prime" was not coined until much later, in the 20th century, by other mathematicians who expanded upon Wilson's original theorem. The study of Wilson primes is a subset of number theory, a branch of pure mathematics dedicated to the study of the integers.
Properties
Wilson primes are a subset of prime numbers, which are integers greater than one that have no divisors other than one and themselves. However, Wilson primes are distinguished by their unique property, which is derived from Wilson's Theorem. This theorem states that for any prime number p, the factorial of (p-1), when increased by one, is divisible by p. In the case of Wilson primes, this result is divisible by p squared.
There are only three known Wilson primes: 5, 13, and 563. Despite extensive computational searches, no other Wilson primes have been found, leading to the conjecture that there may be infinitely many Wilson primes. However, this remains an open question in number theory.
Computational Searches for Wilson Primes
Due to the rarity of Wilson primes, extensive computational searches have been conducted to find more of these elusive numbers. The largest known Wilson prime is 563, discovered in the 20th century. Despite the use of powerful computers and sophisticated algorithms, no other Wilson primes have been found. This has led to the conjecture that there may be infinitely many Wilson primes, although this remains unproven.
Significance in Number Theory
The study of Wilson primes has significant implications in number theory, particularly in the field of prime number theory. The rarity of Wilson primes and the computational difficulty in finding them highlight the complexity and unpredictability of prime numbers. The open question of whether there are infinitely many Wilson primes also contributes to the ongoing exploration of the distribution and properties of prime numbers.