Let $m,n\geq 1$. What is $$\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?$$ Is it $\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?

Let $m,n\geq 1$. What is $$\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?$$ 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This was solved independently by Hegyvári [He25] and Cambie (unpublished), who show that if $m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 $s$.