Phi Function Proof

**wolfamdeus** · May 21st 2009, 03:08 PM

Show that there are infinitely many integers n for which $\phi(n)$ is a perfect square. Hint: Consider $n = 2^{2k + 1}$

I'm trying a proof by induction but I get stuck when considering the k + 1th term.
Thanks in advance!

**Media_Man** · May 21st 2009, 05:12 PM

When p is prime, $\phi(p^a)=p^a(1-\frac{1}{p})$ . Therefore, $\phi(2^{2k+1})=p^{2k+1}(1-\frac{1}{2})=p^{2k}=(p^k)^2$

QED

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