Irrationality of reciprocal products in Erdős Problem 1051
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For an increasing sequence of positive integers aₙ satisfying liminf aₙ^(1/2ⁿ)>1, must the convergent series Σₙ1/(aₙaₙ₊₁) be irrational? This is the original growth-condition formulation of Erdős Problem 1051.
Why it matters and what counts as progress
Why it matters: The growth condition asks when a structured reciprocal series is forced to escape rational values.
What counts as progress: Prove irrationality throughout the stated growth range or weaken the growth condition without introducing rational examples.
How the problem developed
- 1980 · Open question [1]
Paul Erdős and Ronald Graham Recorded the reciprocal-product irrationality question under the growth condition liminf aₙ^(1/2ⁿ)>1; Erdős later described a rapid-growth case as known and asked for the strongest such theorem.
Tony Feng and collaborators Reported Aletheia’s solution for the original liminf growth condition; the paper distinguishes this solution from the stronger later golden-ratio criterion.
AI contribution
Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems
For increasing positive integers aₙ with liminf aₙ^(1/2ⁿ)>1, Aletheia’s accepted proof shows that Σₙ 1/(aₙaₙ₊₁) is irrational. This is the original Erdős #1051 statement.
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