Mills' constant is irrational

Abstract

Let $\lfloor x\rfloor$ denote the integer part of $x$. In 1947, Mills constructed a real number $\hat{I}¾>1$ such that $\lfloor \hat{I}{¾}^{3^k}\rfloor$ is always a prime number for every positive integer $k$. We define Mills' constant as the smallest real number $\hat{I}¾$ 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.

12 pages