Solve the equation $2φ(x)=x $ for $x\in\mathbb N^+.$

Question

Solve the equation $2φ(x)=x $ for $x\in\mathbb N^+.$

I know $$x=\prod_\limits{i} p_i^{a_1} =p_1^{a_1}\cdot p_2^{a_2}\cdot p_3^{a_3} \ldots p_n^{a_n}$$ $$\phi(x)=x\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right)\left(1-\frac{1}{p_3}\right)\ldots \left(1-\frac{1}{p_n}\right)$$

But still don't know to solve it. Can anyone help me?

Answer 1

Let $x=2^kh$ with $k\in\mathbb Z_{\ge 0}$, $h\in\mathbb Z_{\ge 1}$, $h$ odd.

$2\phi(x)=x$ implies $k\ge 1$, so $\phi(2^kh)=2^{k-1}h$.

Now use the multiplicativity of the $\phi$ function: $2^{k-1}\phi(h)=2^{k-1}h$,

i.e. $\phi(h)=h$, i.e. $h=1$, i.e. $x=2^k$, which is a solution for all $k\in\mathbb Z_{\ge 1}$.

Answer 2

Hint: $\phi(n) = n \prod_{p | n} \left( 1 - \frac{1}{p} \right)$, so if $\phi(n)=n/2$, then $$\prod_{p | n} \left( 1 - \frac{1}{p} \right)=\frac{1}{2}.$$ If $n$ is a power of $2$, what can we conclude?

If $n$ is not a power of $2$, what can we conclude?