PROVED (LEAN)
This has been solved in the affirmative and the proof verified in Lean.
Let $m,n\geq 1$. What is\[\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?\]Can it be arbitrarily large? Is it $\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?
This was solved independently by Hegyvári
[He25] and Cambie (unpublished), who show that if $m>n$ then the set in question has size\[\leq m^{O(1/\log\log m)},\]and that for any integer $s$ there exist infinitely many pairs $(m,n)$ such that the set in question has size $s$.
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This page was last edited 18 November 2025. View history
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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 #443, https://www.erdosproblems.com/443, accessed 2026-04-11
