Definition:Gelfond-Schneider Constant

Definition

The Gelfond-Schneider constant is defined as $2$ raised to the power of the square root of $2$:

$2^{\sqrt 2}$

Its decimal expansion begins:

$2^{\sqrt 2} \approx 2 \cdotp 66514 \, 4$

This sequence is A007507 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Source of Name

This entry was named for Alexander Osipovich Gelfond and Theodor Schneider.

Historical Note

The Gelfond-Schneider constant arises in the context of the $7$th problem of the Hilbert $23$.

It had been proved transcendental in $1930$ by Rodion Osievich Kuzmin.

However, as a result of the work done by Alexander Osipovich Gelfond and Theodor Schneider in $\text {1934}$ – $\text {1935}$, it could be shown to be a trivial application of the Gelfond-Schneider Theorem.

Hence the name of this constant.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2 \cdotp 665 \, 144 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2 \cdotp 66514 \, 4 \ldots$