OPEN
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
\[\#\{ a_i \leq x\} \leq \pi(x)?\]
Erdős says this question was asked 'during [his] lecture at Queens College [by] one member of the audience (perhaps S. Shapiro)'. Such a sequence of $a_i$ is sometimes called a set of Beurling prime numbers.
Beurling conjectured that if the number of reals in $[1,x]$ of the form $\prod a_i^{k_i}$ is $x+o(\log x)$ then the $a_i$ must be the sequence of primes.
