Mills' constant is irrational

2024/04/30 by Saito, Kota · 2 citations
#11J72 #11J81 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.19461

Abstract

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

Cited by

• Transcendency of variants of Mills' constant
• Intervals without primes near an iterated linear recurrence sequence

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