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QUESTION

For what values of $n$ does the equation $\phi(2n)=\phi(n)$ hold?



ANSWER

If gcd$(2,n)=1$ (i.e. $n$ is odd), then as $\phi$ is
multiplicative, $\phi(2n)=\phi(2)\phi(n)=(2-1)\phi(n)=\phi(n)$.
Thus the equation holds for all odd $n$.

If $n$ is even, $n=2^rm$ say, where gcd$(2,m)=1$, then
$\phi(n)]\phi(2^r)\phi(m)=2^r\left(1-\frac{1}{2}\right)
\phi(m)=2^{r-1}\phi(m)$, while
$\phi(2n)=\phi(2^{r+1}m)=\phi(2^{r+1})\phi(m)=2^r\phi(m)$. Thus
$\phi(n)\neq\phi(2n)$ if $n$ is even. Hence $\phi(n)=\phi(2n)$ if
and only if $n$ is odd.




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