# Gelfond–Schneider constant

Not to be confused with Gelfond's constant.

The Gelfond–Schneider constant or Hilbert number[1] is two to the power of the square root of two:

22 = 2.6651441426902251886502972498731...

which was proved to be a transcendental number by Rodion Kuzmin in 1930.[2] In 1934, Aleksandr Gelfond proved the more general Gelfond–Schneider theorem,[3] which solved the part of Hilbert's seventh problem described below.

## Properties

The square root of the Gelfond–Schneider constant is the transcendental number

${\displaystyle {\sqrt {2^{\sqrt {2}}}}={\sqrt {2}}^{\sqrt {2}}=1.6325269\ldots }$.

This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either 22 is rational, which proves the theorem, or it is irrational (as it turns out to be), and then

${\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}=\left({\sqrt {2}}\right)^{\left({\sqrt {2}}{\sqrt {2}}\right)}=\left({\sqrt {2}}\right)^{2}=2}$

is an irrational to an irrational power that is rational, which proves the theorem.[4][5] The proof is not constructive, as it does not say which of the two cases is true, but it is much simpler than Kuzmin's proof.

## Hilbert's seventh problem

Part of the seventh of Hilbert's twenty three problems posed in 1900 was to prove (or find a counterexample to the claim) that ab is always transcendental for algebraic a ≠ 0, 1 and irrational algebraic b. In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 22.

In 1919, he gave a lecture on number theory and spoke of three conjectures: the Riemann hypothesis, Fermat's Last Theorem, and the transcendence of 22. He mentioned to the audience that he didn't expect anyone in the hall to live long enough to see a proof of this final result.[6] But the proof of this number's transcendence was published by Kuzmin in 1930,[2] well within Hilbert's own lifetime. Namely, Kuzmin proved the case where the exponent b is a real quadratic irrational, which was later extended to an arbitrary algebraic irrational b by Gelfond.