---
hash: 7b483243751e5437
url: "https://www.erdosproblems.com/951"
final_url: "https://www.erdosproblems.com/951"
family: generic
title: "951 | Erdős Problems"
method: direct
fetched_at: "2026-09-18T08:44:58Z"
sha256_md: 227066a3b8bb31d583b17717bd92b278bdefa1175e1dd0209c30470a2db98129
cited_by:
  - "2050888883294916926"
---
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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](https://www.erdosproblems.com/951): [Er69,p.82][Er77c,p.68]

[number theory](https://www.erdosproblems.com/tags/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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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:  T. F. Bloom, Erdős Problem #951, https://www.erdosproblems.com/951, accessed 2026-09-18

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Formalised statement?  [Yes](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/951.lean)

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