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Erdős Problem #1000

PROVED LEANdatabase updated 2025-12-27

The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.

What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).

Given an infinite sequence of positive integers \(A = \{n_1 < n_2 < \cdots\}\), `phiSeq n k` is \(\phi_A(k)\): the number of \(1\leq m\leq n_k\) such that \(\frac{m}{n_k}\) cannot be written as \(\frac{b}{n_j}\) for any integer \(b\) and any \(1\leq j<k\); equivalently, \[ \frac{n_k}{(m,n_k)}\nmid n_j \] for all \(1\leq j<k\).

The average \(\frac{1}{N}\sum_{k\leq N}\frac{\phi_A(k)}{n_k}\).

Let \(A=\{n_1<n_2<\cdots\}\) be an infinite sequence of positive integers, and let \(\phi_A(k)\) count the number of \(1\leq m\leq n_k\) such that the fraction \(\frac{m}{n_k}\) cannot be written as \(\frac{b}{n_j}\) for any integer \(b\) and any \(j<k\); equivalently, \[ \frac{n_k}{(m,n_k)}\nmid n_j \] for all \(1\leq j<k\). Is there a sequence \(A\) such that \[ \lim_{N\to \infty}\frac{1}{N}\sum_{k\leq N}\frac{\phi_A(k)}{n_k}=0? \] This was solved by Haight [Ha] who proved that such a sequence does exist (contrary to Erdős' expectations).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1000.
Prizenone
Topicsnumber theory, diophantine approximation
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1000

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