Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\ge 0$, have density $>0$?

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

Let [ R_C:={,n\in\mathbb N:\ n=p+\lfloor C^k\rfloor\ \text{for some prime }p\text{ and }k\ge 0,}. ] Interpreting “density” in the usual Erdős/Romanoff sense as **positive lower asymptotic density** [ \underline d(R_C):=\liminf_{x\to\infty}\frac{|R_C\cap[1,x]|}{x}>0, ] the current state of knowledge is:

llm-hunter · gpt pro 5.2 · unverified

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