SOLVED
Are there infinitely many $n$ such that $\tau(n)=\tau(n+1)$, where $\tau$ is the divisor function?
A problem of Erdős and Mirsky
[ErMi52]. More generally, they ask about the estimation of the longest run of consecutive integers $\leq x$ which have the same number of divisors.
Spiro [Sp81] proved that there are infinitely many $n$ such that $\tau(n)=\tau(n+5040)$, and Heath-Brown [He84] improved her method to prove this for $\tau(n)=\tau(n+1)$. More generally, he showed that the number of such $n\leq x$ is
\[\gg \frac{x}{(\log x)^7}.\]
