Problem 852

Erdős Problem #852: declared status 'open'. Formalized: no. Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n<x$ the numbers $d_n,d_{n+1},\ldots,d_{n+h(x)-1}$ are all distinct. Estimate $h(x)$. In particular, is it true that$$h(x) >(\log x)^c$$for some constant $c>0$, and$$h(x)=o(\log x)?$$ Prize: no. OEIS: A001223, A053597, A078515. Tags: number theory, primes.