Let $A=\{\sum {ϵ}_k{3}^k:{ϵ}_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\{\sum {ϵ}_k{4}^k:{ϵ}_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $4$. Does $A+B$ have positive lower density?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

If you write [ C=A+B=\Big{\sum \epsilon_k3^k+\sum \eta_j4^j:\epsilon_k,\eta_j\in{0,1}\Big}, ] then the question is whether (C) has **positive (lower) asymptotic density**, i.e. whether [ \liminf_{x\to\infty}\frac{|C\cap[1,x]|}{x}>0. ] This is an old problem of Burr–Erdős–Graham–Li / Erdős, and it is still listed as ope…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.