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OP-8915OpenMathematics · general

Erdős #269 · irrationality

Let PP be a finite set of primes with ∣P∣≥2\lvert P\rvert \geq 2 and let {a1<a2<⋯ }={n∈N:if p∣n then p∈P}\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}. Is the sum

\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]}, $$…
Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let PP be a finite set of primes with ∣P∣≥2\lvert P\rvert \geq 2 and let {a1<a2<⋯ }={n∈N:if p∣n then p∈P}\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}. Is the sum

∑n=1∞1[a1,…,an],\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},

where [a1,…,an][a_1,\ldots,a_n] is the lowest common multiple of a1,…,ana_1,\ldots,a_n, irrational?

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Formal statement

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Background

If PP is infinite this sum is always irrational (in [Er88c] Erdős says this is a 'simple exercise').

This problem was asked by Erdős in a letter to the editor written January 1st 1973 in issue 12 of the Fibonacci Quarterly, 1974, p. 335. In that letter he says that he can prove the sum is irrational if duplicate summands are removed.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #269 link
  2. [2]Upstream Lean 4 statement for Erdős #269 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 269 link

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