Erdős (Erdos) Problem #114: status in 2026

Erdős Problem #114

If p(z) ∈ ℂ[z] is a monic polynomial of degree n, is the length of the curve z ∈ ℂ : lvert p(z)rvert = 1 maximised when p(z) = z^n - 1? We require n ≥ 1. For n = 0 the only monic polynomial is 1, whose level set is all of ℂ, while z^0 - 1 = 0 has an empty level set.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-12-28). It is marked falsifiable there: if it is false, a finite computation can show it. Erdős offered a prize of $250.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #114 as falsifiable (status updated 28 December 2025).
Formal Conjectures marks the Lean statement erdos_114 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #114 on erdosproblems.com	falsifiable 2025-12-28	erdosproblems.com
Formal Conjectures: status of `erdos_114`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 2 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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-114,
  title        = {Erdős Problem #114},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-114}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

If $p (z) \in C [z]$ is a monic polynomial of degree $n$ , is the length of the curve ${z \in C : ∣ p (z)∣ = 1}$ maximised when $p (z) = z^{n} - 1$ ?

We require $n \geq 1$ . For $n = 0$ the only monic polynomial is $1$ , whose level set is all of $C$ , while $z^{0} - 1 = 0$ has an empty level set.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«114».

theorem erdos_114 :
    ∀ n : ℕ, 1 ≤ n → ∀ p : ℂ[X], p.Monic → p.natDegree = n →
      lemniscateLength p ≤ lemniscateLength (X ^ n - 1)

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/114. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/114
[EHP58] Erdős, P., Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
[ErHa99] Eremenko, A. and Hayman, W., On the length of lemniscates. Michigan Math. J. (1999), 409-415. arXiv:0805.2295
[Ta25] Tao, T., The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree (2025). arXiv:2512.12455

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.