Erdős Problem #481

The problem

Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by

$$(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}.$$

Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.

Statement quoted from erdosproblems.com/481, maintained by Thomas Bloom. Original source: [ErGr80,p.96]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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