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I want to know if it's proofed, that every number which is in the number sequence of the powers of $2$ has an $\phi$ of $\frac12x$.

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2 Answers 2

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Yes, if $p$ is a prime number then $\phi(p^x)=p^{x-1}(p-1)$, so for $p=2$ we have $\phi(2^x)=2^{x-1}(2-1)=2^{x-1}=\frac{1}{2}\times 2^x$.

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  • $\begingroup$ Accepted user133281s answer cause every number you can divide by 2 has a phi of exactly its half. $\endgroup$
    – jankal
    Feb 1, 2015 at 7:52
  • $\begingroup$ That is not true, since in general an odd number does not have to be coprime to an even number. For instance $\phi(6)=2$, because we do not count $3$. $\endgroup$
    – user133281
    Feb 1, 2015 at 10:14
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If $x$ is a power of $2$, an element $y$ of $\{1,2,\ldots,x\}$ is coprime with $x$ iff $y$ is odd. Exactly half of the elements of $\{1,2,\ldots,x\}$ are odd ...

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