Erdős Problem #125 — Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\{ \sum\epsilon_k4^k : \epsilon_k\in \{0,1\}\}$ be the se…

Burr, Erdős, Graham, and Li posed the question in 1996: whether the sumset of base-$3$ and base-$4$ numbers using only digits $0$ and $1$ has positive lower density. The equivalent formulation asks for a constant $c>0$ such that $\lvert(A+B)\cap[1,x]\rvert\geq cx$ for all sufficiently large $x$.

Known results

• Melfi (2001): $\lvert(A+B)\cap[1,x]\rvert\gg x^{0.965}$.
• Hasler and Melfi (2024): improved this to $\gg x^{0.9777}$.
• Hasler and Melfi (2024): upper-bounded the lower density by $1015/1458\approx0.69616$.

May 2026 claimed resolution

A DeepMind-generated inductive thinning argument using approximations $3^m\approx4^k$ claims that the lower density is $0$. The argument was formalized in Lean, but no independently established proof or peer-reviewed confirmation was found.

Current status (as of May 2026): The conjecture is claimed false, with lower density $0$, but the claim remains unverified independently.