Let $k\ge 2$ be large and let $S(k)$ be the minimal $x$ such that there is a positive density set of $n$ where$$n+1,n+2,\dots ,n+k$$are all divisible by primes $\le x$.Estimate $S(k)$ - in particular, is it true that $S(k)\ge k^{1-o(1)}$?

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

I’m going to read “$n+i$ is divisible by primes (\le x)” as: **each** (n+1,\dots,n+k) has **at least one** prime divisor (\le x) [[nomath]](equivalently, none of them is coprime to the primorial $\prod_{p\le x}p$)[[/nomath]]. Under that interpretation, your $S(k)$ is essentially the inverse of the **primorial Jacobstha…

llm-hunter · gpt pro 5.2 · unverified

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