Problem Statement
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 $f(n)/\log n\to\infty$?
Categories:
Number Theory
Progress
Investigated by Erdős and Turán [ErTu34] (prompted by a question of Lázár and Grünwald) in their first joint paper, where they proved that\[\log n \ll f(n) \ll n/\log n\](the upper bound is trivial, taking $A=\{1,\ldots,n\}$). Erdős says that $f(n)=o(n/\log n)$ has never been proved, but perhaps never seriously attacked.This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
Source: erdosproblems.com/126 | Last verified: January 13, 2026