  In 2010, Matom\"{a}ki investigated the set of $A>1$ such that the integer
part of $ A^{c^k} $ is a prime number for every $k\in \mathbb{N}$, where $c\geq
2$ is any fixed real number. She proved that the set is uncountable, nowhere
dense, and has Lebesgue measure $0$. In this article, we show that the set has
Hausdorff dimension $1$.
