OPEN
This is open, and cannot be resolved with a finite computation.
What is the size of the largest $A\subseteq \{1,\ldots,n\}$ such that if $a\leq b\leq c\leq d\in A$ are such that $abcd$ is a square then $ad=bc$?
The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
A question of Erdős, Sárközy, and Sós. Erdős claims that Sárközy proved that $\lvert A\rvert =o(n)$ (a proof of this bound is provided by Tao in the comments).
The primes show that $\lvert A\rvert \gg n/\log n$ is possible. Cambie and Weisenberg have noted in the comments that the set of semiprimes also works, showing\[(1+o(1))\frac{\log\log n}{\log n}n \leq \lvert A\rvert\]is achievable.
See also
[121].
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This page was last edited 22 January 2026. View history
Additional thanks to: Stijn Cambie, Bhavik Mehta, Terence Tao, and Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #888, https://www.erdosproblems.com/888, accessed 2026-05-21
