Erdős Problem #148 — open problem in number theory

Erdős Problem #148

Let F(k) be the number of solutions to 1= 1/n_1+⋯+1/n_k, where 1≤ n_1<⋯<n_k are distinct integers. Find good estimates for F(k).

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-148,
  title        = {Erdős Problem #148},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-148}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $F (k)$ be the number of solutions to $1 = \frac{1}{n _{1}} + \dots + \frac{1}{n _{k}},$ where $1 \leq n_{1} < \dots < n_{k}$ are distinct integers. Find good estimates for $F (k)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«148». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_148 : (fun k ↦ (F k : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/148. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/148
[ElPl21] Elsholtz, Christian and Planitzer, Stefan, *Sums of four and more unit fractions and approximate parametrizations*. Bull. Lond. Math. Soc. (2021), 695-709.
[Ko14] Konyagin, S. V., *Double exponential lower bound for the number of representations of unity by Egyptian fractions*. Math. Notes (2014), 277-281.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.