Mills' constant is irrational

Kota Saito

doi:10.1112/mtk.70027

Mills' constant is irrational

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Abstract

Let (Formula presented.) denote the integer part of (Formula presented.). In 1947, Mills constructed a real number (Formula presented.) such that (Formula presented.) is always a prime number for every positive integer (Formula presented.). We define Mills' constant as the smallest real number (Formula presented.) 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.

Original language	English
Article number	e70027
Journal	Mathematika
Volume	71
Issue number	3
DOIs	https://doi.org/10.1112/mtk.70027
Publication status	Published - Jul 2025

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abstract = "Let (Formula presented.) denote the integer part of (Formula presented.). In 1947, Mills constructed a real number (Formula presented.) such that (Formula presented.) is always a prime number for every positive integer (Formula presented.). We define Mills' constant as the smallest real number (Formula presented.) 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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N2 - Let (Formula presented.) denote the integer part of (Formula presented.). In 1947, Mills constructed a real number (Formula presented.) such that (Formula presented.) is always a prime number for every positive integer (Formula presented.). We define Mills' constant as the smallest real number (Formula presented.) 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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Mills' constant is irrational

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