Mills' constant is irrational

TL;DR

This paper proves that Mills' constant, a number associated with a prime-generating property, is irrational and provides partial results on its transcendence, addressing a long-standing mathematical question.

Contribution

The paper establishes the irrationality of Mills' constant and offers initial insights into its transcendental nature, advancing understanding of this mathematical constant.

Findings

Mills' constant is proven to be irrational.

Partial results on the transcendence of Mills' constant are obtained.

Addresses a long-standing open problem in number theory.

Abstract

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.