Problem 126

Erdős Problem #126: declared status 'open'. Formalized: yes. Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\log n\to\infty$? Current best: Investigated by Erd\H{o}s and Tur\'{a}n [ErTu34] (prompted by a question of L\'{a}z\'{a}r and Gr\"{u}nwald) in their first joint paper, where they proved that$$\log n \ll f(n) \ll n/\log n$$(the upper bound is trivial, taking $A=\{1,\ldots,n\}$). Prize: $250. Tags: number theory.