Mills' constant is irrational

Let $\lfloor x \rfloor$ denote the integer part of $x$. In 1947, Mills constructed a real number $\xi > 1$ such that $\lfloor \xi^{3^k} \rfloor$ is always a prime number for every positive integer $k$. We define Mills' constant as the smallest real number $\xi$ satisfying this property. Determining whether this number is irrational has been a long-standing problem. In this paper, we show that Mills' constant is irrational. Furthermore, we obtain partial results on the transcendency of this number.