Introducing Wilson's Theorem: Uses & Resources

[matqkks]

What is the most motivating way to introduce Wilson’s Theorem? Why is Wilson’s theorem useful? With Fermat’s little Theorem we can say that working with residue 1 modulo prime p makes life easier but apart from working with a particular (p-1) factorial of a prime what other reasons are there for Wilson’s theorem to be useful?
Are there any good resources on this topic?

 

[jvicens]

I would like to start by saying that Wilson's Theorem is a powerful and elegant result in number theory, which has numerous applications in mathematics and other fields. It states that if a number p is prime, then (p-1)! + 1 is divisible by p. This result was discovered by the mathematician John Wilson in 1770 and has been used extensively in various areas of mathematics.

One of the most motivating ways to introduce Wilson's Theorem is by first discussing the concept of primes and their properties. This can be followed by a discussion on how Wilson's Theorem is a generalization of Fermat's Little Theorem. This will help students understand the importance and significance of Wilson's Theorem.

One of the main reasons why Wilson's Theorem is useful is its connection to the concept of primality testing. This theorem provides a simple and efficient way to determine if a number is prime or composite. This is especially useful in cryptography and coding theory, where prime numbers play a crucial role.

Moreover, Wilson's Theorem has applications in combinatorics and number theory, such as in the study of permutations and combinations. It also has connections to other areas of mathematics, such as graph theory and algebraic structures.

In addition to these, Wilson's Theorem has also been used in various real-world applications, such as in the design of efficient algorithms and in determining the number of possible combinations in certain scenarios.

As for resources on this topic, there are many excellent textbooks and online resources available that explain Wilson's Theorem in detail and provide examples of its applications. Some recommended resources include "Elementary Number Theory" by David M. Burton and "An Introduction to the Theory of Numbers" by G.H. Hardy and E.M. Wright.

In conclusion, Wilson's Theorem is a beautiful result in number theory that has numerous applications in mathematics and other fields. Its importance and usefulness make it a fundamental concept that should be introduced and studied in depth.