Omega Constant is Transcendental
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Theorem
The omega constant is transcendental.
Proof
From the definition of omega constant, it is the real number $\Omega$ such that:
- $\Omega \, e^\Omega = 1$
where $e$ denotes Euler's number.
Aiming for a contradiction, suppose $\Omega$ is not transcendental.
Hence, by definition, $\Omega$ is algebraic.
Then $e^\Omega$ is also algebraic, because:
- $e^\Omega = \dfrac 1 \Omega$
However, by the weaker Hermite-Lindemann-Weierstrass theorem, $e^\Omega$ is transcendental.
From this contradiction it follows that $\Omega$ is transcendental.
$\blacksquare$