For any $n$ let $F (n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $ϕ (m + 1), \dots, ϕ (m + k)$ with $m + k \leq n$ . Is it true that $F (n) = (c + o (1)) lo g lo g lo g n$ for some constant $c$ ? Is the first pattern which fails to appear always $ϕ (m + 1) > ϕ (m + 2) > \dots > ϕ (m + k)?$ Is it true that the 'natural' ordering which mimics what happens to $ϕ (1), \dots, ϕ (k)$ is the most likely to appear?

Worked, still open.

number theory · open · possible · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

Let me separate what is known rigorously from what is conjectural, and also flag one definitional issue.

llm-hunter · gpt pro 5.2 · unverified

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

sketched · paper

open