Problem Statement
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?
Categories:
Number Theory
Progress
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.
Source: erdosproblems.com/1004 | Last verified: January 19, 2026