Problem Statement
Let $A\subset \mathbb{N}$ be an additive basis of order $2$. Must there exist $B=\{b_1<b_2<\cdots\}\subseteq A$ which is also a basis such that\[\lim_{k\to \infty}\frac{b_k}{k^2}\]does not exist?
Categories:
Number Theory Additive Basis
Progress
Erdős originally asked whether this was true with $A=B$, but this was disproved by Cassels [Ca57].This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
Source: erdosproblems.com/326 | Last verified: January 14, 2026