OPEN
This is open, and cannot be resolved with a finite computation.
Is\[\sum_{n\geq 2}\frac{1}{n!-1}\]irrational?
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The decimal expansion is
A331373 in the OEIS. Weisenberg has observed that this sum can also be written as\[\sum_{k\geq 1}\sum_{n\geq 2}\frac{1}{(n!)^k}.\]Erdős
[Er88c] notes that $\sum \frac{1}{n!+t}$ should be transcendental for every integer $t$.
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This page was last edited 28 September 2025. View history
Additional thanks to: Desmond Weisenberg
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 #68, https://www.erdosproblems.com/68, accessed 2026-08-10
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