Problem 266

Let $a_n$ be an infinite sequence of positive integers such that $\sum \frac{1}{a_n}$ converges. There exists some integer $t\geq 1$ such that

\sum \frac{1}{a_n+t}

is irrational.

Source: erdosproblems.com/266

Irrationality

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Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved: Kovač and Tao's 2024 construction of a sequence whose shifted reciprocal sums are rational for every rational shift is recorded on its claim page (Kovač and Tao, 2024). The site (page last edited 2025-09-28) labels the problem DISPROVED (LEAN) and credits them with the negative answer; the Lean proof behind the label is a public formalization of their argument linked from the claim page, not built or audited in this corpus.

Problem 266

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Proof claims

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