OPEN
This is open, and cannot be resolved with a finite computation.
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\[h(x)=o(\log x)?\]
Brun's sieve implies $h(x) \to \infty$ as $x\to \infty$.
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #852, https://www.erdosproblems.com/852, accessed 2025-12-07
