## A Formula for φ

We would like to develop a formula for Euler’s φ function so that we can apply Euler’s Theorem in practice. This video states simply what we are trying to accomplish, which is to look at two special cases.

## General Formulas for φ(p^{2}) and φ(pq)

This video explores a derivation for the special cases φ(p^{2}) and φ(pq), and compares our results for our two formulas.

## A Visual Representation of Our Proof

Another way of understanding our proof for φ(p^{2}) and φ(pq) is to represent how we counted the multiples of p and q with a Venn diagram. We will have some work to do in order to derive a more general result for φ; we will need to take a slight detour through the Chinese Remainder Theorem in order to arrive at a general expression.