Problem #220: Let $n\geq 1$ and\[A=\{a_1<\cdots

Let $n\geq 1$ and\[A=\{a_1<\cdots

SOLVED $500

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Progress

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].

This is discussed in problem B40 of Guy's collection [Gu04].