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Erdős #1041 · analysis / polynomials

Let f(z)=∏i=1n(z−zi)∈C[z]f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z] with ∣zi∣<1\lvert z_i\rvert < 1 for all ii.

Must there always exist a path of length less than 22 in

{z:∣f(z)∣<1}\{z: \lvert f(z)\rvert < 1\}

which connects two of the roots o…

Source reports FALSIFIABLEAs of 2026-09-08Source & evidence ↓
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Problem statement

Let f(z)=∏i=1n(z−zi)∈C[z]f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z] with ∣zi∣<1\lvert z_i\rvert < 1 for all ii.

Must there always exist a path of length less than 22 in

{z:∣f(z)∣<1}\{z: \lvert f(z)\rvert < 1\}

which connects two of the roots of ff?

Lean 4

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Formal statement

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Background

A problem of Erdős, Herzog, and Piranian [EHP58], who proved that this set always has a connected component containing at least two of the roots.

Progress & source assessment

The source labels this entry FALSIFIABLE in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1041 link
  2. [2]Upstream Lean 4 statement for Erdős #1041 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1041 link

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