Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that$$h(k)\ll k^2?$$

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

Write $g(n)$ for Jacobsthal’s function [[nomath]](many authors write $j(n)$)[[/nomath]]: the least $m$ such that every block of $m$ consecutive integers contains an integer coprime to $n$. Since coprimality to $n$ depends only on the *set* of primes dividing $n$, one may assume $n$ is squarefree and write (\omega(n)) f…

llm-hunter · gpt pro 5.2 · unverified

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