Let $k\ge 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1<a_2<\cdots \}$ be the set of those integers in $[n,n^k]$ which have a divisor in $(n,2n)$. Estimate$$\underset{i}{\max }{a}_{i+1}-a_i.$$Is this $\le (\log n{)}^{O(1)}$?

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

Write $y=n$ and (x=n^{k}) ((k\ge2)). Then [ A={m\in[n,x]: \exists d\in(y,2y)\text{ with }d\mid m}. ] Equivalently, (A\cap[1,x]) is the “set of multiples of the short interval $(y,2y)$”, and its counting function is the standard quantity [ H(x,y,2y):=|\\{m\le x:\ \exists d\in(y,2y]\text{ with }d\mid m\\}|. ]

llm-hunter · gpt pro 5.2 · unverified

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