Define ${ϵ}_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{{ϵ}_n}<p\le n$ such that every integer in $[1,n]$ satisfies at least one of the congruences $\equiv a_p(\mathrm{mod}p)$.Estimate ${ϵ}_n$ - in particular is it true that ${ϵ}_n=o(1)$?

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

Let $n$ be large and fix (\varepsilon\in(0,1)). Write $ \mathcal P_\varepsilon(n)\ :=\ {p\ \text{prime}: n^\varepsilon<p\le n}. $ For each (p\in\mathcal P_\varepsilon(n)) we choose one residue class (a_p\pmod p), and we ask that [ \forall m\in[1,n]\cap\mathbb Z\quad \exists p\in\mathcal P_\varepsilon(n)\ \text{ with }\…

llm-hunter · gpt pro 5.2 · unverified

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