The board / number theory / #46
Erdős Problem #46
PROVED LEANdatabase updated 2025-12-29
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).
Does every finite colouring of the integers have a monochromatic solution to \(1=\sum \frac{1}{n_i}\) with \(2\leq n_1<\cdots <n_k\)? The answer is yes, as proved by Croot [Cr03] - indeed, there are infinitely many disjoint such monochromatic solutions.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/46.Prizenone
Topicsnumber theory, unit fractions, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/46
Attempts
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