# euler phi-function

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• Mar 29th 2009, 11:00 AM
htata123
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
• Mar 29th 2009, 11:10 AM
ThePerfectHacker
Quote:

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.
• Mar 29th 2009, 11:13 AM
htata123
I did that but I'm still stuck. Any ideas on how I should proceed?
• Mar 29th 2009, 07:42 PM
ThePerfectHacker
Quote:

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.