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Level A · Machine-checkable Hard Number theory P-erdos-252

Erdős Problem #252

Erdős Problem 252: irrationality of the sum for a given 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-252,
  title        = {Erdős Problem #252},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-252}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
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Disputed
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Erdős Problem 252: irrationality of the sum for a given .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«252». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_252 :
    answer(sorry) ↔ ∀ k ≥ 1, Irrational (erdos_252_sum k)

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/252. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_252.variants.k_ge_five — For a fixed k ≥ 5, is ∑ σ k n / n! irrational?.

References

  • erdosproblems.com/252
  • [ErSt71] Erdös, P., and E. G. Straus. "Some number theoretic results." Pacific J. Math 36 (1971): 635-646.
  • [ErSt74] Erdős, Paul, and Ernst Straus. "On the irrationality of certain series." Pacific journal of mathematics 55.1 (1974): 85-92.
  • [ErKa54] P. Erdős, M. Kac, Amer. Math. Monthly 61 (1954), Problem 4518.
  • [ScPu06] Schlage-Puchta, J. C., The irrationality of a number theoretical series. Ramanujan J. (2006), 455-460.
  • [FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers (2007).
  • [Pr22] Pratt, K., The irrationality of a divisor function series of Erdős and Kac. arXiv:2209.11124 (2022).

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.