erdos_1003.variants.eps87

Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$. [EPS87] Erd\H os, Paul and Pomerance, Carl and S\'ark\"ozy, Andr\'as, _On locally repeated values of certain arithmetic functions_. {II}. Proc. Amer. Math. Soc. (1987), 1--7.

theorem erdos_1003.variants.eps87 :  ∀ᶠ x in atTop,
    {(n : ℕ) | (n ≤ x) ∧ φ n = φ (n + 1)}.ncard ≤
      x / Real.exp ((x.log) ^ ((1 : ℝ) / 3))

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