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Let $F(x)$ be a function of the form $ \sum_{i=1}^r d_i x^{\rho_i}$ where$d_1,\ldots,d_r\in\mathbb{R}$, $0 \leq \rho_1 < \ldots < \rho_r,$ $\rho_r\not\in \mathbb{Z},\rho_i \in \mathbb{R}$ for $ 1 \leq i \leq r$ and$d_r\not=0$. We prove that sets of the form $\{ n \in \mathbb{N}: \{ F(n) \}\in U \}$ for any non-empty open set $U \subset [0,1)$ contain arbitrarily longstrings of consecutive primes.

Arbitrarily long strings of consecutive primes in special sets