Problem #402: Prove that, for any finite set $A\subset\mathbb{N}$, there...

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Progress

A conjecture of Graham [Gr70], who also conjectured that (assuming $A$ itself has no common divisor) the only cases where equality is achieved are when $A=\{1,\ldots,n\}$ or $\{L/1,\ldots,L/n\}$ (where $L=\mathrm{lcm}(1,\ldots,n)$) or $\{2,3,4,6\}$.

Proved for all sufficiently large sets (including the sharper version which characterises the case of equality) independently by Szegedy [Sz86] and Zaharescu [Za87].

Proved for all sets by Balasubramanian and Soundararajan [BaSo96].