A Couple of Transcendental Prime-Representing Constants

TL;DR

This paper proves certain prime-representing constants are transcendental using advanced approximation theorems, clarifying their algebraic nature.

Contribution

It introduces methods to establish the transcendence of specific prime-representing constants, advancing understanding of their mathematical properties.

Findings

Certain prime-representing constants are proven transcendental.

Liouville and Roth's theorems are effectively applied to these constants.

The algebraic nature of Mills' constant remains uncertain.

Abstract

It is well known that the arithmetic nature of Mills' prime-representing constant is uncertain: we do not know if Mills' constant is a rational or irrational number. In the case of other prime-representing constants, irrationality can be proved, but it is not known whether these constants are algebraic or transcendental numbers. By using Liouville or Roth's theorems about approximation by rationals, we find a couple of prime-representing constants that can be proved to be transcendental numbers.