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 $p$.