# Math Help - euler phi-function

1. ## euler phi-function

Hi,

I need help with this proof:

Prove that there are no integers n for which phi(n) = n/4

Appreciate any assistance

2. Originally Posted by htata123
Hi,

I need help with this proof:

Prove that there are no integers n for which phi(n) = n/4

Appreciate any assistance
Write $n$ in a prime factorization and use formula for the phi-function.

3. I did that but I'm still stuck. Any ideas on how I should proceed?

4. Originally Posted by htata123
I did that but I'm still stuck. Any ideas on how I should proceed?
Certainly $n$ is divisible by $4$ so we can write $n=2^a\cdot m$ where $a\geq 2$ and $m$ is a positive odd integer. Now, $\phi(n) = \phi(2^a)\phi(m) = 2^{a-1}\phi(m)$ since $(2^a,m)=1$. While $n/4 = 2^{a-2}m$. Therefore, if $\phi(n) = n/4$ then $2^{a-1}\phi(m) = 2^{a-2}m\implies 2\phi(m) = m$. This is impossible because LHS is even while RHS is odd.