The board / number theory / #1190
Erdős Problem #1190
SOLVED LEANdatabase updated 2026-05-14
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. 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).
\(\epsilon_m\) is the supremum of \(\sum \frac{1}{n_i}\) over all finite sequences \(m<n_1<\cdots<n_k\) for which there exist congruences \(a_i\pmod{n_i}\) such that no integer satisfies two such congruences.
Let \[\epsilon_m=\max \sum \frac{1}{n_i}\] where the maximum is taken over all finite sequences \(m<n_1<\cdots<n_k\) for which there exist congruences \(a_i\pmod{n_i}\) such that no integer satisfies two such congruences. Estimate \(\epsilon_m\). The resolution of [202] by GPT-5.4 Pro implies via the same reduction that \[\epsilon_m=L(m)^{-1+o(1)},\] where \(L(m)=\exp(\sqrt{\log m\log\log m})\).
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/1190.Prizenone
Topicsnumber theory, covering systems
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1190
Attempts
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