Submit progress

on Upper bound on the maximal number of distinct prime factors of pairwise sums

The statement

For each positive integer $n$ , let $f(n)$ denote the largest integer $m$ such that for every set $A \subseteq \mathbb{N}$ with $|A| = n$ , the product $\prod_{\substack{a, b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Determine whether $f(n) = o\!\left(\frac{n}{\log n}\right)$ , that is, whether $\frac{f(n)}{n / \log n} \to 0$ as $n \to \infty$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.