Gelfond's Constant is Transcendental
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Theorem
- $e^\pi$
is transcendental.
Proof
From the Gelfond-Schneider Theorem:
If:
- $\alpha$ and $\beta$ are algebraic numbers such that $\alpha \notin \set {0, 1}$
- $\beta$ is either irrational or not wholly real
then $\alpha^\beta$ is transcendental.
We have that:
\(\ds i^{-2 i}\) | \(=\) | \(\ds \paren {e^{i \pi / 2} }^{- 2 i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^{-\pi i^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^\pi\) |
As:
- $i$ is algebraic
- $-2 i$ is algebraic and not wholly real
the conditions of the Gelfond-Schneider Theorem are fulfilled.
Hence the result.
$\blacksquare$
Historical Note
The question of the transcendental nature of Gelfond's constant $e^\pi$ was raised in the context of the $7$th problem of the Hilbert $23$.
That Gelfond's Constant is Transcendental was initially established in $1929$ by Alexander Osipovich Gelfond.
It was since determined to be a special case of the Gelfond-Schneider Theorem, established $\text {1934}$ – $\text {1935}$.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $23 \cdotp 140 \, 692 \, 632 \, 7792 \, 69 \, 005 \, 729 \, 086 \ldots$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $23 \cdotp 14069 \, 26327 \, 79269 \, 00572 \, 9086 \ldots$