The board / number theory / #444
Erdős Problem #444
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
\(d_A(n)\) is the number of \(a \in A\) which divide \(n\). Only the positive divisors of \(n\) are counted, so \(0 \in A\) does not change \(d_A(n)\) for \(n \ge 1\).
Let \(A\subseteq\mathbb{N}\) be infinite and \(d_A(n)\) count the number of \(a\in A\) which divide \(n\). Is it true that, for every \(k\), \[\limsup_{x\to \infty} \frac{\max_{n<x}d_A(n)}{\left(\sum_{n\in A\cap[1,x)}\frac{1}{n}\right)^k} =\infty?\] This is asked in [ErGr80, p.88]. The answer is yes, proved by Erdős and Sárközy [ErSa80]. The maximum is over \(1 \le n < x\), since every \(a \in A\) divides \(0\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/444.Prizenone
Topicsnumber theory, divisors
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/444
Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.