One of the neatest things about \(\phi(n)\), beyond being quite useful for things we are familiar with (congruences), is that it is a prototype for the many functions there are in number theory. So we will look at it in a bit more depth.

Let's get some more conjectures about values of \(\phi(n)\). Finding patterns is fun!

We already saw Euler's Theorem, that if \(\gcd(a,n)=1\), then \(a^{\phi(n)}\equiv 1\) mod (\(n\)).

But there are some other places one might look for patterns, now that one has done some number theory. These are questions the Fundamental Theorem of Arithmetic just begs us to ask.

• Given a prime \(p\), is there a formula for \(\phi(p^e)\)?
• If \(m\) and \(n\) are coprime, is there a relation between \(\phi(mn)\) and \(\phi(m)\) and \(\phi(n)\)?

There are a lot of other interesting questions one can ask about this function. For instance, one can ask:

• When does \(\phi(n)\mid n\)?
• Given \(m\), for how many integers \(n\) it is true that \(\phi(n)=m\)?
• Are there infinitely many \(n\) for which \(\phi(n)\) ends in zero?

Quite surprisingly, there is an additive result as well — try \[\sum_{d\mid n} \phi(d)\] to seek a pattern!

Finally, one can ask questions about new, related functions. For instance, let \(f(n)=\phi(n)/n\). Can you find a formula? Where is this function equal to certain values, such as \(f(n)=1/2\)?

Before moving on to some proofs in the next section, we highly encourage all readers to explore a lot of this. It's simply not the same to just prove; one must get a feel for these things to really understand them.