Erdős Problem #517

If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?

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-517,
  title        = {Erdős Problem #517},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-517}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Disputed

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On the literature board

The problem

The question

If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«517». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_517 : answer(sorry) ↔ ∀ {f : ℂ → ℂ} {n : ℕ → ℕ} (hn : HasFabryGaps n)
    {a : ℕ → ℂ} (ha : ∀ k, a k ≠ 0) (hf : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (z : ℂ),
    {x : ℂ | f x = z}.Infinite

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/517. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/517
[Bi28] Biernacki, Miécislas, Sur les équations algébriques contenant des paramétres arbitraires. (1928), 145.

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.