The board / number theory / #25
Erdős Problem #25
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(n_1 < n_2 < \dots\) be an arbitrary sequence of integers, each with an associated residue class \(a_i \pmod{n_i}\). Let \(A\) be the set of integers \(n\) such that for every \(i\) either \(n < n_i\) or \(n \not\equiv a_i \pmod{n_i}\). Must the logarithmic density of \(A\) exist?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/25.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/25
Attempts
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Proved that bounded pairwise gcds force an explicit natural density, so every counterexample must have unbounded common factors; verified 27,567 finite systems and an exact nonsummable gcd-2 example.
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