Problem 951

Erdős Problem #951: declared status 'open'. Formalized: yes. 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$$\#\{ a_i \leq x\} \leq \pi(x)?$$ Prize: no. Tags: number theory.