Erdős Problem #265

declared status 'open'. Formalized: no. How fast can $a_n\to \infty$ grow if$$\sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n-1}$$are both rational? Current best: If we replace $-1$ by a different constant then higher degree polynomials can be used - for example if we consider $\sum_{n\geq 2}\frac{1}{a_n}$ and $\sum_{n\geq 2}\frac{1}{a_n-12}$ then $a_n=n^3+6n^2+5n$ is an example of both series being rational. Prize: no. Tags: irrationality.

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