Problem 432

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Statement. Let $A,B\subseteq \mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?

Status. Open, the site's label (OPEN). A comment of 23 June 2026 on the site's discussion thread links Sungchul Lee's manuscript On the Density of Pairwise Coprime Sumsets (GitHub). The author used OpenAI's GPT-5.5 Pro to explore proof strategies. Write $S(x)=|(A+B)\cap[1,x]|$. The manuscript proves $S(x)\le\pi(x)$ whenever the distinct elements of $A+B$ are pairwise coprime. It constructs infinite $A,B$ with pairwise coprime sums and $S(x)\gg(\log x/\log\log x)^2$ for all large $x$. For every $F(x)=x^{o(1)}$, it constructs infinite $A,B$ with $S(x_j)\ge F(x_j)$ along a sequence $x_j\to\infty$. Assuming the Hardy--Littlewood prime-tuples conjecture, for every $\omega(x)\to\infty$ it gives $A,B$ whose sums are distinct primes, with $S(x_j)\ge x_j/(\log x_j)^{\omega(x_j)}$ along a sequence. These bounds settle no instance of the question, so the manuscript has no claim page.

Source. erdosproblems.com/432, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #432, https://www.erdosproblems.com/432.

Formalization. None recorded.

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