Erdős Problem #229

Let $(S_n)_{n\geq 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n\geq 1$, there exists some $k_n\geq 0$ such that\[f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?\]

#229: [Er57][Er61,p.250][Ha74][Er82e]

analysis | iterated functions

This is Problem 2.30 in [Ha74], where it is attributed to Erdős.

Solved in the affirmative by Barth and Schneider [BaSc72].

This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.

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This page was last edited 29 December 2025.

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Additional thanks to: AlphaProof and team, Zachary Chase, and Terence Tao

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #229, https://www.erdosproblems.com/229, accessed 2026-02-13