Problem Statement
Estimate the maximum of $F(A,B)$ as $A,B$ range over all subsets of $\{1,\ldots,N\}$, where $F(A,B)$ counts the number of $m$ such that $m=ab$ has exactly one solution (with $a\in A$ and $b\in B$).
Categories:
Number Theory
Progress
In the comments van Doorn proves\[(1+o(1))\frac{N^2}{\log N}\leq \max_{A,B}F(A,B) \ll \frac{N^2}{(\log N)^\delta(\log\log N)^{3/2}}\]where $\delta=1-\frac{1+\log\log 2}{\log 2}\approx 0.086$.See also [490].
Source: erdosproblems.com/896 | Last verified: January 19, 2026