Gelfond–Schneider constant

The Gelfond–Schneider constant is :$2^\{sqrt\{2=2.6651441...$which was proved by Rodion Kuzmin to be a transcendental number.
Aleksandr Gelfond in 1934 proved the more general "Gelfond–Schneider theorem", which completely solved the part of Hilbert's seventh problem described below.

Its square root is the transcendental number :$sqrt\{2\}^\{sqrt\{2=1.6325269...$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.

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 [David Hilbert, "Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919-1920".] . But the proof of this number's transcendence was published in 1934 [Aleksandr Gelfond, "Sur le septième Problème de Hilbert," Bull. Acad. Sci. URSS Leningrade 7, pp.623-634, 1934.] , well within Hilbert's own lifetime.

The paper by Kuzmin proved the case where the exponent b is a real quadratic number.

ee also

* Gelfond's constant
* Hilbert number

References

* R. Kuzmin, "On a new class of transcendental numbers", Izvestiya Akademii Nauk SSSR, Ser. matem., 7 (1930), 585-597.