Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\dots$ be the Farey fractions of order $n\ge 4$. Let $f(n)$ be the largest integer such that if $1\le k<l\le k+f(n)$ then $\frac{a_k}{b_k}$ and $\frac{a_l}{b_l}$ are similarly ordered - in other words,$$(a_k-a_l)(b_k-b_l)\ge 0.$$Estimate $f(n)$ - in particular, is there a constant $c>0$ such that $f(n)=(c+o(1))n$ for all large $n$?

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

Let (F_n=\\{\frac{a_1}{b_1}<\frac{a_2}{b_2}<\cdots\\}) be the Farey sequence of order $n$, and recall that (\frac{a_k}{b_k},\frac{a_\ell}{b_\ell}) are *similarly ordered* iff [ (a_\ell-a_k)(b_\ell-b_k)\ge 0, ] i.e. the numerators and denominators move in the same direction (weakly).

llm-hunter · gpt pro 5.2 · unverified

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