SOLVED - $500
Let $n\geq 1$ and
\[A=\{a_1<\cdots <a_{\phi(n)}\}=\{ 1\leq m<n : (m,n)=1\}.\]
Is it true that
\[ \sum_{1\leq k<\phi(n)}(a_{k+1}-a_k)^2 \ll \frac{n^2}{\phi(n)}?\]
The answer is yes, as proved by Montgomery and Vaughan
[MoVa86], who in fact proved that for any $\gamma\geq 1$
\[ \sum_{1\leq k<\phi(n)}(a_{k+1}-a_k)^\gamma \ll \frac{n^\gamma}{\phi(n)^{\gamma-1}}.\]
This general form was also asked by Erdős in
[Er73].
