Let $ϵ>0$ and $\omega (n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that$$\omega (n-k)<(1+ϵ)\frac{\log k}{\log \log k}$$for all $k<n$ which are sufficiently large depending on $ϵ$ only?Can one show the stronger version with$$\omega (n-k)<\frac{\log k}{\log \log k}+O(1)$$is false?

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

As far as I can tell, your first question [[nomath]](the $(1+\varepsilon)$-version)[[/nomath]] is **currently open**. It is recorded as Erdős Problems #679 and is listed there as unsolved. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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