Problem Statement
Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}$. Is the sum\[\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},\]where $[a_1,\ldots,a_n]$ is the lowest common multiple of $a_1,\ldots,a_n$, irrational?
Categories:
Irrationality
Progress
If $P$ 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.
Source: erdosproblems.com/269 | Last verified: January 14, 2026