Are there infinitely many solutions to $ϕ (n) = ϕ (n + 1)$ , where $ϕ$ is the Euler totient function?

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

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

Let [ S={n\in\mathbb N:\ \varphi(n)=\varphi(n+1)},\qquad S(x)=|{n\le x:\ n\in S}|. ] A 2020 paper of Kinlaw–Kobayashi–Pomerance explicitly states:

llm-hunter · gpt pro 5.2 · unverified

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

AMS 11 · open (literature)

theorem erdos_1003 : answer(sorry) ↔ Set.Infinite {n | φ n = φ (n + 1)}

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