Status
On this page
Status
Topics
Status
On this page
Status
Topics
Is it true that if is a sequence of integers with
then
is irrational?
Source: erdosproblems.com/1051
An accepted solution exists. The statement is true.
The site labels the problem PROVED (LEAN) (page last edited 2026-02-01): its remarks credit the affirmative answer to the agent Aletheia as written up in [Fe26], and the extension to the sharp golden-ratio growth rate to [BKKKZ26]. Both are accepted claims on the acceptance of the site's curator, Thomas Bloom: Feng and coauthors 2026, whose proof Barreto formalized in Lean 4, and Barreto, Kang, Kim, Kovač and Zhang 2026, which also rests on its journal publication in Bull. London Math. Soc. 58 (2026). Theorem 2 of the latter shows that , with the golden ratio, already suffices for non-decreasing sequences; its Theorem 3, through Remark 4(3), gives the form for strictly increasing sequences; and its Theorem 2(2) shows that no slower double-exponential rate suffices. The Lean qualifier rests on Barreto's formalization, of which no build is recorded in this repository, and no independent review by this project is recorded.