OPEN
This is open, and cannot be resolved with a finite computation.
Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r<n<p_{r+1}$ all of whose prime factors are $<p_{r+1}-p_r$.
Erdős thought this was true but that there are very few such $r$. He could show that the density of $r$ such that at least one such $n$ exist is $0$.
This problem has been
formalised in Lean as part of the
Google DeepMind Formal Conjectures project.
View the LaTeX source
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 #932, https://www.erdosproblems.com/932, accessed 2026-01-16
