PROVED (LEAN)
This has been solved in the affirmative and the proof verified in Lean.
Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $\sum_{k\geq 1}\frac{1}{u_k+1}$ and $u_k=\lfloor c_0^{2^k}+1\rfloor$ for $k\geq 1$, where\[c_0=\lim u_n^{1/2^n}=1.264085\cdots.\]Let $a_1<a_2<\cdots $ be any other sequence with $\sum \frac{1}{a_k}=1$. Is it true that\[\liminf a_n^{1/2^n}<c_0=1.264085\cdots?\]
An earlier interpretation of this question on this site defined $u_1=2$ and $u_{n+1}=u_n^2-u_n+1$ (
Sylvester's sequence), which is the same sequence shifted by $1$; we use the phrasing above as more faithful to
[ErGr80].
The constant $c_0$ is called the Vardi constant.
This is true, and was proved independently by Kamio
[Ka25] and Li and Tang
[LiTa25].
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This page was last edited 01 February 2026. View history
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Additional thanks to: Quanyu Tang and Wouter van Doorn
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 #315, https://www.erdosproblems.com/315, accessed 2026-04-11
