951 | Erdős Problems

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OPEN This is open, and cannot be resolved with a finite computation.

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\[\#\{ a_i \leq x\} \leq \pi(x)?\]

#951: [Er69,p.82][Er77c,p.68]

number theory

Erdős

[Er77c]

said this question was asked 'during [his] lecture at Queens College [by] one member of the audience (perhaps S. Shapiro)'. (Although in

[Er80]

he seems sure it was Shapiro, but also recalled that he had himself asked this in

[Er69]

.) In

[Er80]

he further asks whether equality holds if and only if the $a_i$ are the set of primes.

Such a sequence of $a_i$ is sometimes called a set of Beurling prime numbers (and the sequence of products called the associated generalised integers).

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.

It is unclear whether this is intended to hold for all $x$ or just all sufficiently large $x$. The former question can be disproved by finite calculation (since any finite sequence of $a_i$ can be extended to a suitable infinite sequence via a greedy algorithm). A finite counterexample for $x=10$ was found by ChatGPT-5.2 Pro (prompted by Leeham).

Additional thanks to: Kevin Barreto, Leeham, and Terence Tao

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