Problem 1004

Erdős Problem #1004: declared status 'open'. Formalized: yes. 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? Current best: Erd\H{o}s, Pomerenace, and S\'{a}rk\"{o}zy [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 $c>0$. Prize: no. Tags: number theory.