Wilson's theorem

The set of Wilson ( named after John Wilson ) is a mathematical theorem in number theory. It makes Teilbarkeitsaussagen to natural or whole numbers and is therefore also associated with the basic theory of numbers, with the methods, it can be also demonstrated.

Set

The set of Wilson is: Let be a natural number. Then just then a prime number if divisible by. It refers to the faculty, so the product.

With the help of the concept of congruence can be the set also formulated as follows: Let be a natural number, it shall

Conversely, one can conclude with the sentence also: Let be a natural number, it shall

So is not divisible by, so is a prime number. But is divisible by, we obtain from the set of Wilson the information that is assembled without knowing a specific factorization. However, the computational effort for the faculty is not less than sample divisions.

Examples

The following table shows the values ​​of n of from 2 to 30, ( n-1)! and the rest of (n -1)! modulo n if n is a prime number, then the background color is pink. And if n is a composite number, then the background color is light green.

History

This is known today as a set of Wilson result was first discovered by Ibn al - Haytham, but eventually named after John Wilson ( a student of the English mathematician Edward Waring ), who rediscovered it more than 700 years later. Waring published this theorem in 1770, although neither he nor Wilson could provide proof. Lagrange gave the first proof in 1773. There is reason to believe that Leibniz a century ago knew of this result, but it never published.

Generalizations

It is generally:

A slight generalization of the Wilson is:

A number is prime if and only if for all

Applies. This set is easy to prove by induction by and with the set of Wilson. For and yields the set of Wilson. If, here, the result is:

With and is odd if and only prime if.

Related terms

Primes, where even is divisible by hot Wilson primes.