Erdős Problem #1051

Is it true that if \(a_0 < a_1 < a_2 < \cdots\) is a strictly increasing sequence of integers with \(\liminf a_n^{1/2^n} > 1\), then the series \(\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}\) is irrational? This was solved in the affirmative by Aletheia [Fe26]. This was extended by Barreto, Kang, Kim, Kovač, and Zhang [BKKKZ26], who essentially give a complete answer: if \(\phi=\frac{1+\sqrt{5}}{2}\) is the golden ratio and \(1\leq a_1 < a_2 < \cdots\) is a monotonically increasing sequence of integers such that \(\limsup a_n^{1/\phi^{n}}=\infty\) then \(\sum_{n=1}^\infty \frac{1}{a_na_{n+1}}\) is irrational. Conversely, for any \(1 < C < \infty\) there exists a sequence of integers \(1\leq a_1<\cdots\) such that \(\lim a_n^{1/\phi^{n}}=C\) where this infinite sum is a rational number. (Further, more general, results are available in [BKKKZ26].) This was formalized in Lean by Baretto.