Problem 1003

Erdős Problem #1003: declared status 'open'. Formalized: yes. Are there infinitely many solutions to $\phi(n)=\phi(n+1)$, where $\phi$ is the Euler totient function? Current best: Erd\H{o}s [Er85e] says that, presumably, for every $k\geq 1$ the equation$$\phi(n)=\phi(n+1)=\cdots=\phi(n+k)$$has infinitely many solutions. Erd\H{o}s, Pomerance, and S\'{a}rk\"{o}zy [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. Prize: no. OEIS: A001274. Tags: number theory.