Note On The Catalan Constant And Prime Triples

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 $β(2)=0.9159655941\ldots$ is an irrational number. This note provides the detailed analysis. Moreover, the numerical data suggests that the irrationality measure is $μ(β(2))=2$, the same as almost every irrational real numbers.

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