Erdős Problem #264
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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A sequence \(a_n\) of integers is called an irrationality sequence if for every bounded sequence of integers \(b_n\) with \(a_n + b_n \neq 0\) and \(b_n \neq 0\) for all \(n\), the sum \[ \sum \frac{1}{a_n + b_n} \] is irrational. Note: there are other possible definitions of this concept. See FormalConjectures/ErdosProblems/263.lean for another possible definition.
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Proved fixed-bound factorial-tail separation and Hausdorff dimension zero, reduced a positive factorial answer to open problem #68, and exactly certified that every rational sum for all admissible $\{
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