- #1
sutupidmath
- 1,630
- 4
Homework Statement
Well, the question is to find a formula for [tex]\phi(p^n)[/tex] where [tex]\phi[/tex] is the Euler's totient, and p is a positive prime.
Homework Equations
[tex]\phi(2^n)[/tex] well in this case here is what i did, i first took n=1, n=2, n=3, and it looks like
[tex]\phi(2^n)=2^{n-1}[/tex] but i am failing to prove it, because this is just by observation.
Also i tried
[tex]\phi(3^n)[/tex] and by observing what happens when, n=1,2,3 etc, this one also looks like the general formula for any n is going to be
[tex]\phi(3^n)=2*3^{n-1}[/tex] again i wasn't able to prove this one formally either.
I tried in both cases to prove it by induction, but i have a feeling that i need to know more properties about that function in order to know what operations i can perform. We, basically have only learned what euler's function is, and that's it.
The Attempt at a Solution
I really don't know whot to go about it.
Well, i gave it a shot, but nothing
say n=1 then
[tex]\phi(p)=p-1[/tex] since the group of invertibles is going to have p-1 elements.
Now, let n=2
[tex]\phi(p^2)=?[/tex]