OPEN
This is open, and cannot be resolved with a finite computation.
Let $c>0$. If $x$ is sufficiently large then does there exist $n\leq x$ such that the values of $\phi(n+k)$ are all distinct for $1\leq k\leq (\log x)^c$, where $\phi$ is the Euler totient function?
Erdős, Pomerenace, and Sárközy
[EPS87] proved that if $\phi(n+k)$ are all distinct for $1\leq k\leq K$ then\[K \leq \frac{n}{\exp(c(\log n)^{1/3})}\]for some constant $c>0$.
See
[945] for the analogous problem with the divisor function.
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T. F. Bloom, Erdős Problem #1004, https://www.erdosproblems.com/1004, accessed 2026-01-16
