# Gelfond's Constant is Transcendental

## Contents

## Theorem

- $e^\pi$

is transcendental.

## Proof

From the Gelfond-Schneider Theorem:

If:

- $\alpha$ and $\beta$ are algebraic numbers such that $\alpha \notin \left\{{0, 1}\right\}$
- $\beta$ is either irrational or not wholly real

then $\alpha^\beta$ is transcendental.

We have that:

\(\displaystyle i^{- 2 i}\) | \(=\) | \(\displaystyle \left({e^{i \pi / 2} }\right)^{- 2 i}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle e^{- \pi i^2}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle e^\pi\) | $\quad$ | $\quad$ |

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 $1934$ – $1935$.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($1809$ – $1882$) - 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): $23 \cdotp 14069 \, 26327 \, 79269 \, 00572 \, 9086 \ldots$