The board / number theory / #275
Erdős Problem #275
PROVED LEANdatabase updated 2026-01-20
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).
If a finite system of \(r\) congruences \(\{ a_i\pmod{n_i} : 1\leq i\leq r\}\) (the \(n_i\) are not necessarily distinct) covers \(2^r\) consecutive integers then it covers all integers. This is best possible as the system \(2^{i-1}\pmod{2^i}\) shows. This was proved independently by Selfridge and Crittenden and Vanden Eynden [CrVE70]. This was formalized in Lean by Alexeev using Aristotle.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/275.Prizenone
Topicsnumber theory, covering systems
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/275
Attempts
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