Note On The Catalan Constant And Prime Triples

TL;DR

This paper explores the connection between the distribution of prime triples and the irrationality of the Catalan constant, providing detailed analysis and numerical evidence supporting its irrationality measure.

Contribution

It establishes a link between prime triples and the irrationality of the Catalan constant, offering detailed analysis and numerical data to support the conjecture.

Findings

Infinitely many prime triples imply the Catalan constant is irrational.

Numerical data suggests the irrationality measure of the Catalan constant is 2.

Detailed analysis supports the conjecture of the Catalan constant's irrationality.

Abstract

The existence of infinitely many consecutive prime triples $p_n$, $ p_{n+1}$, and $p_{n+2}$ as $n \to \infty$, is sufficient to prove that the Catalan constant $\beta(2)=0.9159655941\ldots $ is an irrational number. This note provides the detailed analysis. Moreover, the numerical data suggests that the irrationality measure is $\mu(\beta(2))=2$, the same as almost every irrational real numbers.