Problem Statement
Let $u_1=2$ and $u_{n+1}=u_n^2-u_n+1$. Let $a_1<a_2<\cdots $ be any other sequence with $\sum \frac{1}{a_k}=1$. Is it true that\[\liminf a_n^{1/2^n}<\lim u_n^{1/2^n}=c_0=1.264085\cdots?\]
Categories:
Number Theory Unit Fractions
Progress
$c_0$ is called the Vardi constant and the sequence $u_n$ is Sylvester's sequence.In [ErGr80] this problem is stated with the sequence $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, but Quanyu Tang has pointed out this is probably an error (since with that choice we do not have $\sum \frac{1}{u_n}=1$). This question with Sylvester's sequence is the most natural interpretation of what they meant to ask.
This is true, and was proved independently by Kamio [Ka25] and Li and Tang [LiTa25].
Source: erdosproblems.com/315 | Last verified: January 14, 2026