Problem 1210

../

---

Statement. Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$. Is it true that

$$\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p<n}\frac{1}{p}+O(1)?$$

Status. Open.

Source. erdosproblems.com/1210, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1210, https://www.erdosproblems.com/1210.

References.

• [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation). The statement above, labeled OPEN on the site and glossed as not resolvable by a finite computation. The site's commentary, in this page's words: in [Er80] Erdős says he stated the problem incorrectly in [Er77c], and the [Er77c] problem he presumably means concerns the primes $n<q_1<\cdots<q_k\le m$ of $(n,m]$, asking whether $\sum 1/(q_i-n)$ is less than $\sum_{p<m-n}1/p+O(1)$. The site links Problems 460 and 950. The formal-conjectures file states both the main question (erdos_1210) and that prime-interval form (erdos_1210.variants.er80_correction), each as an open research problem with no proof.

Progress. No result settling the question, any class of sets $A$, or any range of $n$ is recorded. On 8 April 2026 the site's curator relayed a suggestion from GPT Pro that the bound should follow quickly from standard sieve results, then added an edit doubting it, since $A$ may contain many primes just below $n$. A reply the same day, crediting GPT-5.4 Thinking, showed that the estimate the route needs would imply $\pi(x+y)\le\pi(x)+\pi(y)+O(y/(\log y)^2)$, an unproved weak form of the inequality of Problem 855, so standard results do not settle the problem by that route.

Recorded reason: the Bado note. I. O. Bado, "A Dyadic-Farey Reduction for an Erdős Problem on Pairwise Coprime Sets", ResearchGate preprint, https://doi.org/10.13140/RG.2.2.33043.03361, registered 2026-05-24, was linked in the thread on 31 May 2026 as claiming some results on the problem but no full solution. Its DataCite and OpenAlex records carry no abstract, and no theorem from it is recorded, so it has no claim page.

Search scope (2026-10-07). The site page and its forum thread, the formal-conjectures file, and the DataCite and OpenAlex records of the Bado note.

Gaps. The question is open; the corpus holds no claim on it. The content of the Bado note is unrecorded, and a theorem from it settling any case would need its own claim page.