OPEN
This is open, and cannot be resolved with a finite computation.
Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that\[\lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1\]and $\sum\frac{1}{a_n}\in \mathbb{Q}$. Then, for all sufficiently large $n\geq 1$,\[ a_n = a_{n-1}^2-a_{n-1}+1.\]
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Erdős and Straus
[ErSt64] proved that if $\lim a_n/a_{n-1}^2=1$ and $\sum \frac{1}{a_n}$ is rational, and $a_n$ does not satisfy the recurrence, then\[\limsup_{n\to \infty} \frac{[a_1,\ldots,a_n]}{a_{n+1}}\left(\frac{a_n^2}{a_{n+1}}-1\right)>0.\]A sequence satisfying the reucrrence $a_n = a_{n-1}^2-a_{n-1}+1$ is known as
Sylvester's sequence.
Duverney
[Du01] proved a weaker version of this problem: if\[\sum_{n\geq 0}\left(\frac{a_{n+1}}{a_n^2}-1\right)\]converges then $\sum \frac{1}{a_n}$ is rational if and only if\[a_{n}=a_{n-1}^2-a_{n-1}+1\]for all large $n$.
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This page was last edited 21 January 2026. View history
External data from
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Formalised statement?
Yes
Related OEIS sequences:
A000058
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Additional thanks to: Alfaiz
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 #243, https://www.erdosproblems.com/243, accessed 2026-04-11
