Gelfond–Schneider constant

The Gelfond–Schneider constant or Hilbert number<ref>Template:Citation</ref> is two to the power of the square root of two:

2^{Template:Sqrt} ≈ Template:Val...

which was proved to be a transcendental number by Rodion Kuzmin in 1930.<ref name=Kuzmin>Template:Cite journal</ref> In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general Gelfond–Schneider theorem,<ref>Template:Cite journal</ref> which solved the part of Hilbert's seventh problem described below.

Properties

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

<math>\sqrt{2^{\sqrt{2}}}=\sqrt{2}^{\sqrt{2}} \approx </math> Template:Val....

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 <math>\sqrt{2}^\sqrt{2}</math> is a rational which proves the theorem, or it is irrational (as it turns out to be) and then

<math>\left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}=\sqrt{2}^{\sqrt{2}\times\sqrt{2}}=\sqrt{2}^2=2</math>

is an irrational to an irrational power that is a rational which proves the theorem.<ref>Template:Citation.</ref><ref>Template:Citation,</ref> 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

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Part of the seventh of Hilbert's twenty-three problems posed in 1900 was to prove, or find a counterexample to, the claim that a^b 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 2^{Template:Sqrt}.

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 2^{Template:Sqrt}. He mentioned to the audience that he didn't expect anyone in the hall to live long enough to see a proof of this result.<ref>David Hilbert, Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919–1920.</ref> But the proof of this number's transcendence was published by Kuzmin in 1930,<ref name=Kuzmin/> 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 and by Schneider.

See also

• Gelfond's constant

Further reading

• Template:Cite book
• Template:Cite book