Problem #1003: Are there infinitely many solutions to $\phi(n)=\phi(n+1)$,...

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Progress

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.

Erdős, Pomerance, and Sárközy [EPS87] proved that the number of $n\leq x$ with $\phi(n)=\phi(n+1)$ is at most\[\frac{x}{\exp((\log x)^{1/3})}.\]See [946] for the analogous question with the divisor function.