The board / number theory / #467
Erdős Problem #467
ambiguous statement
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The finite set of primes at most \(x\).
The sets \(A\) and \(B\) form a disjoint nonempty decomposition of the primes up to \(x\).
The selected congruence classes cover every \(n<x\) from both sides of the partition.
For every sufficiently large \(x\), choose a congruence class \(a_p\pmod p\) for each prime \(p\leq x\). Can the primes be decomposed into two nonempty sets \(A\) and \(B\) such that every \(n<x\) lies in a chosen class from \(A\) and in a chosen class from \(B\)? This follows the interpretation on the Erdős Problems website, which notes that the original source omits some crucial quantifiers.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/467.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/467
Attempts
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Exact anchor/knapsack certificates rule out every x<=7012 under 1<=n<x; 7013 is only the first relaxation admission, and the asymptotic two-colour problem remains open.
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