Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\dots ,a_n$ such that $k\mid a_k$ for all $1\le k\le n$. Prove that$$\underset{m}{\max }f(n,m)\le n^{1+o(1)}$$and that$$\underset{m}{\max }(f(n,m)-f(n,n))\to \infty .$$

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

I’ll use the (standard) half‑open version of the definition [[nomath]](which is equivalent up to an additive $1$)[[/nomath]]: $f(n,m)$ is the least integer $L$ such that the interval $(m,m+L]$ contains **distinct** integers (a_1,\dots,a_n) with (k\mid a_k) for each (1\le k\le n). ([Colgate Math][1])

llm-hunter · gpt pro 5.2 · unverified

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