A proof of Wilson's Theorem using polynomial interpolation

In the last posts, we have learned about polynomial interpolation through two formulas, the Newton's interpolation formula and the Lagrange's interpolation formula. Both formulas can be used to prove the Wilson's theorem.

In this post, we will present a proof of the Wilson's theorem based on the Newton's interpolation formula. The other proof, that is, proving Wilson's theorem based on the Lagrange's interpolation formula, we leave it as a homework exercise.

Wilson's Theorem is a well known theorem in number theory. It says that if $p$ is prime number then the number $(p−1)!+1$ is a multiple of $p$.

Lagrange polynomial interpolation

Today we will continue to learn about polynomial interpolation. In the last post, we have learned about Newton's interpolation formula, today we will learn a different interpolation formula called the Lagrange interpolation formula.

We will use the following example $$P(x) = 2x^2 - 3x + 3$$

This polynomial $P(x)$ is a polynomial of degree $2$ and we can calculate that $$P(1) = 2, ~~P(2) = 5, ~~P(3) = 12.$$

The polynomial interpolation problem is the problem of reconstructing the polynomial $P(x)$, given the condition that $P(1) = 2$, $P(2) = 5$, and $P(3) = 12$.

Newton polynomial interpolation

Today, we will learn about polynomial interpolation.

Suppose we have the following polynomial $$P(x) = 2x^2 - 3x + 3$$

Let us calculate the values of the polynomial $P(x)$ for a few values of $x$ as follows $$P(1) = 2 - 3 + 3 = 2,$$ $$P(2) = 8 - 6 + 3 = 5,$$ $$P(3) = 18 - 9 + 3 = 12, \dots$$

Now, suppose that we are given the following information $$P(1) = 2, ~~P(2) = 5, ~~P(3) = 12,$$ can we reconstruct the polynomial $P(x)$?

The answer is, yes, we can. An interpolation formula enables us to reconstruct the polynomial $P(x)$ based on its values. Today, we will learn about Newton's interpolation formula, and in the next post, we will cover Lagrange's interpolation.