Gelfond–Schneider constant

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The Gelfond–Schneider constant is

which Aleksandr Gelfond proved to be a transcendental number using the Gelfond–Schneider theorem, answering one of the questions raised in Hilbert's seventh problem.

Its square root is the transcendental number

which can be used to show that an irrational number to the power of an irrational number can sometimes produce a rational number, since this number raised to the power of √2 is equal to 2.

[edit] 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 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^√2.

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^√2. 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^[1]. But the proof of this number's transcendence was published in 1934^[2], well within Hilbert's own lifetime.

[edit] See also

• Gelfond's constant
• Hilbert number

[edit] References

1. ^ David Hilbert, Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919-1920.
2. ^ Aleksandr Gelfond, Sur le septième Problème de Hilbert, Bull. Acad. Sci. URSS Leningrade 7, pp.623-634, 1934.