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reviewer:erdos-db-trust · no key registered on this bundle

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null → 5144e816

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in state · unreviewed

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1. 2026-05-30proposeproposed · finding.addagent:erdos-spine-ingestvpr_ef2a2364e670ce45Candidate claim vc_f4d6defc70020a0c imported from artifact packet cap_61973ee16b553d57
2. 2026-05-30acceptfinding.assertedreviewer:erdos-db-trustnull→5144e816vev_e906e3267e9339b7Candidate claim vc_f4d6defc70020a0c imported from artifact packet cap_61973ee16b553d57

Erdős Problem #1051 has status 'proved (lean)'. Statement: 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. Topics: irrationality. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.