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arXiv · 2404.19461

Mills' constant is irrational

Kota Saito
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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.

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Kota Saito. 2024-04-30. Mills' constant is irrational. https://doi.org/10.1112/mtk.70027

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