The board / number theory / #308
Erdős Problem #308
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`k` is representable as a sum of distinct unit fractions with denominators from `{1, …, N}`.
The set of positive integers representable as a sum of distinct unit fractions with denominators from `{1, …, N}`.
`f N` is the smallest positive integer which is not representable as a sum of distinct unit fractions with denominators from `{1, …, N}`.
Let \(N\geq 1\). What is the smallest integer not representable as the sum of distinct unit fractions with denominators from \(\{1,\ldots,N\}\)? This was essentially solved by Croot [Cr99] (see `erdos_308.variants.croot`); in particular, for all sufficiently large \(N\), the smallest such integer is either \(m_N\) or \(m_N+1\), where \(m_N=\lfloor \sum_{n\leq N}\frac{1}{n}\rfloor\).
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/308.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA101877
Original pageerdosproblems.com/308
Attempts
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