Problem 299

Is there an infinite sequence $a_1<a_2<\cdots$ such that $a_{i+1}-a_i=O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to $1$ ?

Source: erdosproblems.com/299

Number theory
Unit fractions

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved, in the site's label. Every such bounded-gap sequence has a finite reciprocal sum equal to one, by Bloom's positive-upper-density theorem. The site's label is DISPROVED (LEAN); the linked formalization evidence is specified below.

ErGr80: P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980), p. 36, as cited by the site.
Bl21: T. F. Bloom, On a density conjecture about unit fractions, arXiv:2112.03726 (2021), v2 (2023), Theorem 2; with Appendix B co-written by T. F. Bloom and B. Mehta; JEMS 27 (2025), 4563–4589.
LiSa24: Y. P. Liu and M. Sawhney, On further questions regarding unit fractions, arXiv:2404.07113v1 (10 April 2024), Theorem 1.1.

Problem 299

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