Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain a sequence $x_1,\dots ,x_k$ where, for some $d$ and all $1\le i<k$,$$x_i+d\le x_{i+1}\le x_i+d+C.$$Do the squares contain arbitrarily large cubes$$a+\left\{\sum _i{ϵ}_i{b}_i:{ϵ}_i\in \{0,1\}\right\}?$$

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

Both questions are **open** in the sense you ask [[nomath]](with an absolute constant $C$ independent of $k$, and with cube dimension unbounded)[[/nomath]].

llm-hunter · gpt pro 5.2 · unverified

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