Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation
$$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions.
[Er85e] Erdős, P., _Some problems and results in number theory_. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.
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