Irrationality of rapidly converging series: A problem of Erdős and Graham

2026/01/29 by Kevin Barreto, Jiwon Kang, Sang-hyun Kim +2 · 3 voices · 1 citation
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #Mathematical Dynamics and Fractals #math.CA #math.NT #semigroups and automata theory

paper · pdf · doi:10.1112/blms.70457

arxiv published 2026/01/29 · arxiv updated 2026/07/08 · openalex publication_date 2026/07/29 · openalex created_date 2026/07/30 · openalex updated_date 2026/08/01

Abstract

Abstract Answering a question of Erdős and Graham, we show that the double‐exponential growth condition for a strictly increasing sequence of positive integers is sufficient for the series to have an irrational sum; here denotes the golden ratio. We also provide a positive generalization to , and a negative result showing that some of these instances are essentially optimal. The original problem was autonomously solved by the AI agent Aletheia , powered by Gemini Deep Think, while the remaining material is largely a product of human–AI interactions.

Citations

• Lacunary sequences whose reciprocal sums represent all rational numbers in an interval
• On the irrationality of certain super-polynomially decaying series
• On several irrationality problems for Ahmes series
• On the set of points represented by harmonic subseries
• On the irrationality of Cantor series
• Criteria for Irrationality of Certain Classes of Numbers
• On the irrationality of Cantor and Ahmes series

Cited by

• Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

Discussions

• Most recently this one: arxiv.org/abs/2601.21442 I personally use it in software development a lot. [bsky, 2 points, 1 comments]
• Correction: Agent is called Aletheia in the preprint. It was apparently misspelled in that discussion thread. [bsky, 2 points, 1 comments]
• My collaborators and I later generalised Aletheia’s solution to 1051 to give this paper that I’m quite proud of. I think there’s room to push the results further but we haven’t explored it yet. arxiv. [bsky, 1 points, 0 comments]