Erdős Problem #951

The problem

Let $1<a_1<\cdots$ be a sequence of real numbers such that

$$\left\lvert \prod_i a_i^{k_i}-\prod_j a_j^{\ell_j}\right\rvert \geq 1$$

for every distinct pair of non-negative finitely supported integer tuples $k_i,\ell_j\geq 0$. Is it true that

$$\#\{ a_i \leq x\} \leq \pi(x)?$$

Statement quoted from erdosproblems.com/951, maintained by Thomas Bloom. Original source: [Er69,p.82][Er77c,p.68]. Current status there: open.

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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